Calculus Examples

$f (x) = x^{\frac{5}{11}} {(x - 5)}^{2}$

Step 1

Find the first derivative of the function.

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Step 1.1

Rewrite ${(x - 5)}^{2}$ as $(x - 5) (x - 5)$ .

$\frac{d}{d x} [x^{\frac{5}{11}} ((x - 5) (x - 5))]$

Step 1.2

Expand $(x - 5) (x - 5)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$\frac{d}{d x} [x^{\frac{5}{11}} (x (x - 5) - 5 (x - 5))]$

Step 1.2.2

Apply the distributive property.

$\frac{d}{d x} [x^{\frac{5}{11}} (x \cdot x + x \cdot - 5 - 5 (x - 5))]$

Step 1.2.3

Apply the distributive property.

$\frac{d}{d x} [x^{\frac{5}{11}} (x \cdot x + x \cdot - 5 - 5 x - 5 \cdot - 5)]$

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $x$ by $x$ .

$\frac{d}{d x} [x^{\frac{5}{11}} (x^{2} + x \cdot - 5 - 5 x - 5 \cdot - 5)]$

Step 1.3.1.2

Move $- 5$ to the left of $x$ .

$\frac{d}{d x} [x^{\frac{5}{11}} (x^{2} - 5 \cdot x - 5 x - 5 \cdot - 5)]$

Step 1.3.1.3

Multiply $- 5$ by $- 5$ .

$\frac{d}{d x} [x^{\frac{5}{11}} (x^{2} - 5 x - 5 x + 25)]$

Step 1.3.2

Subtract $5 x$ from $- 5 x$ .

$\frac{d}{d x} [x^{\frac{5}{11}} (x^{2} - 10 x + 25)]$

Step 1.4

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{\frac{5}{11}}$ and $g (x) = x^{2} - 10 x + 25$ .

$x^{\frac{5}{11}} \frac{d}{d x} [x^{2} - 10 x + 25] + (x^{2} - 10 x + 25) \frac{d}{d x} [x^{\frac{5}{11}}]$

Step 1.5

Differentiate.

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Step 1.5.1

By the Sum Rule, the derivative of $x^{2} - 10 x + 25$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 10 x] + \frac{d}{d x} [25]$ .

$x^{\frac{5}{11}} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 10 x] + \frac{d}{d x} [25]) + (x^{2} - 10 x + 25) \frac{d}{d x} [x^{\frac{5}{11}}]$

Step 1.5.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$x^{\frac{5}{11}} (2 x + \frac{d}{d x} [- 10 x] + \frac{d}{d x} [25]) + (x^{2} - 10 x + 25) \frac{d}{d x} [x^{\frac{5}{11}}]$

Step 1.5.3

Since $- 10$ is constant with respect to $x$ , the derivative of $- 10 x$ with respect to $x$ is $- 10 \frac{d}{d x} [x]$ .

$x^{\frac{5}{11}} (2 x - 10 \frac{d}{d x} [x] + \frac{d}{d x} [25]) + (x^{2} - 10 x + 25) \frac{d}{d x} [x^{\frac{5}{11}}]$

Step 1.5.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$x^{\frac{5}{11}} (2 x - 10 \cdot 1 + \frac{d}{d x} [25]) + (x^{2} - 10 x + 25) \frac{d}{d x} [x^{\frac{5}{11}}]$

Step 1.5.5

Multiply $- 10$ by $1$ .

$x^{\frac{5}{11}} (2 x - 10 + \frac{d}{d x} [25]) + (x^{2} - 10 x + 25) \frac{d}{d x} [x^{\frac{5}{11}}]$

Step 1.5.6

Since $25$ is constant with respect to $x$ , the derivative of $25$ with respect to $x$ is $0$ .

$x^{\frac{5}{11}} (2 x - 10 + 0) + (x^{2} - 10 x + 25) \frac{d}{d x} [x^{\frac{5}{11}}]$

Step 1.5.7

Add $2 x - 10$ and $0$ .

$x^{\frac{5}{11}} (2 x - 10) + (x^{2} - 10 x + 25) \frac{d}{d x} [x^{\frac{5}{11}}]$

Step 1.5.8

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{5}{11}$ .

$x^{\frac{5}{11}} (2 x - 10) + (x^{2} - 10 x + 25) (\frac{5}{11} x^{\frac{5}{11} - 1})$

Step 1.6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{11}{11}$ .

$x^{\frac{5}{11}} (2 x - 10) + (x^{2} - 10 x + 25) (\frac{5}{11} x^{\frac{5}{11} - 1 \cdot \frac{11}{11}})$

Step 1.7

Combine $- 1$ and $\frac{11}{11}$ .

$x^{\frac{5}{11}} (2 x - 10) + (x^{2} - 10 x + 25) (\frac{5}{11} x^{\frac{5}{11} + \frac{- 1 \cdot 11}{11}})$

Step 1.8

Combine the numerators over the common denominator.

$x^{\frac{5}{11}} (2 x - 10) + (x^{2} - 10 x + 25) (\frac{5}{11} x^{\frac{5 - 1 \cdot 11}{11}})$

Step 1.9

Simplify the numerator.

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Step 1.9.1

Multiply $- 1$ by $11$ .

$x^{\frac{5}{11}} (2 x - 10) + (x^{2} - 10 x + 25) (\frac{5}{11} x^{\frac{5 - 11}{11}})$

Step 1.9.2

Subtract $11$ from $5$ .

$x^{\frac{5}{11}} (2 x - 10) + (x^{2} - 10 x + 25) (\frac{5}{11} x^{\frac{- 6}{11}})$

Step 1.10

Move the negative in front of the fraction.

$x^{\frac{5}{11}} (2 x - 10) + (x^{2} - 10 x + 25) (\frac{5}{11} x^{- \frac{6}{11}})$

Step 1.11

Combine $\frac{5}{11}$ and $x^{- \frac{6}{11}}$ .

$x^{\frac{5}{11}} (2 x - 10) + (x^{2} - 10 x + 25) \frac{5 x^{- \frac{6}{11}}}{11}$

Step 1.12

Move $x^{- \frac{6}{11}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$x^{\frac{5}{11}} (2 x - 10) + (x^{2} - 10 x + 25) \frac{5}{11 x^{\frac{6}{11}}}$

Step 1.13

Simplify.

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Step 1.13.1

Apply the distributive property.

$x^{\frac{5}{11}} (2 x) + x^{\frac{5}{11}} \cdot - 10 + (x^{2} - 10 x + 25) \frac{5}{11 x^{\frac{6}{11}}}$

Step 1.13.2

Apply the distributive property.

$x^{\frac{5}{11}} (2 x) + x^{\frac{5}{11}} \cdot - 10 + x^{2} \frac{5}{11 x^{\frac{6}{11}}} - 10 x \frac{5}{11 x^{\frac{6}{11}}} + 25 \frac{5}{11 x^{\frac{6}{11}}}$

Step 1.13.3

Combine terms.

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Step 1.13.3.1

Multiply $x^{\frac{5}{11}}$ by $x$ by adding the exponents.

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Step 1.13.3.1.1

Move $x$ .

$x \cdot x^{\frac{5}{11}} \cdot 2 + x^{\frac{5}{11}} \cdot - 10 + x^{2} \frac{5}{11 x^{\frac{6}{11}}} - 10 x \frac{5}{11 x^{\frac{6}{11}}} + 25 \frac{5}{11 x^{\frac{6}{11}}}$

Step 1.13.3.1.2

Multiply $x$ by $x^{\frac{5}{11}}$ .

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Step 1.13.3.1.2.1

Raise $x$ to the power of $1$ .

$x^{1} x^{\frac{5}{11}} \cdot 2 + x^{\frac{5}{11}} \cdot - 10 + x^{2} \frac{5}{11 x^{\frac{6}{11}}} - 10 x \frac{5}{11 x^{\frac{6}{11}}} + 25 \frac{5}{11 x^{\frac{6}{11}}}$

Step 1.13.3.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{1 + \frac{5}{11}} \cdot 2 + x^{\frac{5}{11}} \cdot - 10 + x^{2} \frac{5}{11 x^{\frac{6}{11}}} - 10 x \frac{5}{11 x^{\frac{6}{11}}} + 25 \frac{5}{11 x^{\frac{6}{11}}}$

Step 1.13.3.1.3

Write $1$ as a fraction with a common denominator.

$x^{\frac{11}{11} + \frac{5}{11}} \cdot 2 + x^{\frac{5}{11}} \cdot - 10 + x^{2} \frac{5}{11 x^{\frac{6}{11}}} - 10 x \frac{5}{11 x^{\frac{6}{11}}} + 25 \frac{5}{11 x^{\frac{6}{11}}}$

Step 1.13.3.1.4

Combine the numerators over the common denominator.

$x^{\frac{11 + 5}{11}} \cdot 2 + x^{\frac{5}{11}} \cdot - 10 + x^{2} \frac{5}{11 x^{\frac{6}{11}}} - 10 x \frac{5}{11 x^{\frac{6}{11}}} + 25 \frac{5}{11 x^{\frac{6}{11}}}$

Step 1.13.3.1.5

Add $11$ and $5$ .

$x^{\frac{16}{11}} \cdot 2 + x^{\frac{5}{11}} \cdot - 10 + x^{2} \frac{5}{11 x^{\frac{6}{11}}} - 10 x \frac{5}{11 x^{\frac{6}{11}}} + 25 \frac{5}{11 x^{\frac{6}{11}}}$

Step 1.13.3.2

Move $2$ to the left of $x^{\frac{16}{11}}$ .

$2 \cdot x^{\frac{16}{11}} + x^{\frac{5}{11}} \cdot - 10 + x^{2} \frac{5}{11 x^{\frac{6}{11}}} - 10 x \frac{5}{11 x^{\frac{6}{11}}} + 25 \frac{5}{11 x^{\frac{6}{11}}}$

Step 1.13.3.3

Move $- 10$ to the left of $x^{\frac{5}{11}}$ .

$2 x^{\frac{16}{11}} - 10 \cdot x^{\frac{5}{11}} + x^{2} \frac{5}{11 x^{\frac{6}{11}}} - 10 x \frac{5}{11 x^{\frac{6}{11}}} + 25 \frac{5}{11 x^{\frac{6}{11}}}$

Step 1.13.3.4

Combine $x^{2}$ and $\frac{5}{11 x^{\frac{6}{11}}}$ .

$2 x^{\frac{16}{11}} - 10 x^{\frac{5}{11}} + \frac{x^{2} \cdot 5}{11 x^{\frac{6}{11}}} - 10 x \frac{5}{11 x^{\frac{6}{11}}} + 25 \frac{5}{11 x^{\frac{6}{11}}}$

Step 1.13.3.5

Move $5$ to the left of $x^{2}$ .

$2 x^{\frac{16}{11}} - 10 x^{\frac{5}{11}} + \frac{5 \cdot x^{2}}{11 x^{\frac{6}{11}}} - 10 x \frac{5}{11 x^{\frac{6}{11}}} + 25 \frac{5}{11 x^{\frac{6}{11}}}$

Step 1.13.3.6

Move $x^{\frac{6}{11}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$2 x^{\frac{16}{11}} - 10 x^{\frac{5}{11}} + \frac{5 x^{2} x^{- \frac{6}{11}}}{11} - 10 x \frac{5}{11 x^{\frac{6}{11}}} + 25 \frac{5}{11 x^{\frac{6}{11}}}$

Step 1.13.3.7

Multiply $x^{2}$ by $x^{- \frac{6}{11}}$ by adding the exponents.

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Step 1.13.3.7.1

Move $x^{- \frac{6}{11}}$ .

$2 x^{\frac{16}{11}} - 10 x^{\frac{5}{11}} + \frac{5 (x^{- \frac{6}{11}} x^{2})}{11} - 10 x \frac{5}{11 x^{\frac{6}{11}}} + 25 \frac{5}{11 x^{\frac{6}{11}}}$

Step 1.13.3.7.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 x^{\frac{16}{11}} - 10 x^{\frac{5}{11}} + \frac{5 x^{- \frac{6}{11} + 2}}{11} - 10 x \frac{5}{11 x^{\frac{6}{11}}} + 25 \frac{5}{11 x^{\frac{6}{11}}}$

Step 1.13.3.7.3

To write $2$ as a fraction with a common denominator, multiply by $\frac{11}{11}$ .

$2 x^{\frac{16}{11}} - 10 x^{\frac{5}{11}} + \frac{5 x^{- \frac{6}{11} + 2 \cdot \frac{11}{11}}}{11} - 10 x \frac{5}{11 x^{\frac{6}{11}}} + 25 \frac{5}{11 x^{\frac{6}{11}}}$

Step 1.13.3.7.4

Combine $2$ and $\frac{11}{11}$ .

$2 x^{\frac{16}{11}} - 10 x^{\frac{5}{11}} + \frac{5 x^{- \frac{6}{11} + \frac{2 \cdot 11}{11}}}{11} - 10 x \frac{5}{11 x^{\frac{6}{11}}} + 25 \frac{5}{11 x^{\frac{6}{11}}}$

Step 1.13.3.7.5

Combine the numerators over the common denominator.

$2 x^{\frac{16}{11}} - 10 x^{\frac{5}{11}} + \frac{5 x^{\frac{- 6 + 2 \cdot 11}{11}}}{11} - 10 x \frac{5}{11 x^{\frac{6}{11}}} + 25 \frac{5}{11 x^{\frac{6}{11}}}$

Step 1.13.3.7.6

Simplify the numerator.

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Step 1.13.3.7.6.1

Multiply $2$ by $11$ .

$2 x^{\frac{16}{11}} - 10 x^{\frac{5}{11}} + \frac{5 x^{\frac{- 6 + 22}{11}}}{11} - 10 x \frac{5}{11 x^{\frac{6}{11}}} + 25 \frac{5}{11 x^{\frac{6}{11}}}$

Step 1.13.3.7.6.2

Add $- 6$ and $22$ .

$2 x^{\frac{16}{11}} - 10 x^{\frac{5}{11}} + \frac{5 x^{\frac{16}{11}}}{11} - 10 x \frac{5}{11 x^{\frac{6}{11}}} + 25 \frac{5}{11 x^{\frac{6}{11}}}$

Step 1.13.3.8

Combine $- 10$ and $\frac{5}{11 x^{\frac{6}{11}}}$ .

$2 x^{\frac{16}{11}} - 10 x^{\frac{5}{11}} + \frac{5 x^{\frac{16}{11}}}{11} + x \frac{- 10 \cdot 5}{11 x^{\frac{6}{11}}} + 25 \frac{5}{11 x^{\frac{6}{11}}}$

Step 1.13.3.9

Multiply $- 10$ by $5$ .

$2 x^{\frac{16}{11}} - 10 x^{\frac{5}{11}} + \frac{5 x^{\frac{16}{11}}}{11} + x \frac{- 50}{11 x^{\frac{6}{11}}} + 25 \frac{5}{11 x^{\frac{6}{11}}}$

Step 1.13.3.10

Combine $x$ and $\frac{- 50}{11 x^{\frac{6}{11}}}$ .

$2 x^{\frac{16}{11}} - 10 x^{\frac{5}{11}} + \frac{5 x^{\frac{16}{11}}}{11} + \frac{x \cdot - 50}{11 x^{\frac{6}{11}}} + 25 \frac{5}{11 x^{\frac{6}{11}}}$

Step 1.13.3.11

Move $- 50$ to the left of $x$ .

$2 x^{\frac{16}{11}} - 10 x^{\frac{5}{11}} + \frac{5 x^{\frac{16}{11}}}{11} + \frac{- 50 \cdot x}{11 x^{\frac{6}{11}}} + 25 \frac{5}{11 x^{\frac{6}{11}}}$

Step 1.13.3.12

Move $x^{\frac{6}{11}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$2 x^{\frac{16}{11}} - 10 x^{\frac{5}{11}} + \frac{5 x^{\frac{16}{11}}}{11} + \frac{- 50 x \cdot x^{- \frac{6}{11}}}{11} + 25 \frac{5}{11 x^{\frac{6}{11}}}$

Step 1.13.3.13

Multiply $x$ by $x^{- \frac{6}{11}}$ by adding the exponents.

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Step 1.13.3.13.1

Move $x^{- \frac{6}{11}}$ .

$2 x^{\frac{16}{11}} - 10 x^{\frac{5}{11}} + \frac{5 x^{\frac{16}{11}}}{11} + \frac{- 50 (x^{- \frac{6}{11}} x)}{11} + 25 \frac{5}{11 x^{\frac{6}{11}}}$

Step 1.13.3.13.2

Multiply $x^{- \frac{6}{11}}$ by $x$ .

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Step 1.13.3.13.2.1

Raise $x$ to the power of $1$ .

$2 x^{\frac{16}{11}} - 10 x^{\frac{5}{11}} + \frac{5 x^{\frac{16}{11}}}{11} + \frac{- 50 (x^{- \frac{6}{11}} x^{1})}{11} + 25 \frac{5}{11 x^{\frac{6}{11}}}$

Step 1.13.3.13.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 x^{\frac{16}{11}} - 10 x^{\frac{5}{11}} + \frac{5 x^{\frac{16}{11}}}{11} + \frac{- 50 x^{- \frac{6}{11} + 1}}{11} + 25 \frac{5}{11 x^{\frac{6}{11}}}$

Step 1.13.3.13.3

Write $1$ as a fraction with a common denominator.

$2 x^{\frac{16}{11}} - 10 x^{\frac{5}{11}} + \frac{5 x^{\frac{16}{11}}}{11} + \frac{- 50 x^{- \frac{6}{11} + \frac{11}{11}}}{11} + 25 \frac{5}{11 x^{\frac{6}{11}}}$

Step 1.13.3.13.4

Combine the numerators over the common denominator.

$2 x^{\frac{16}{11}} - 10 x^{\frac{5}{11}} + \frac{5 x^{\frac{16}{11}}}{11} + \frac{- 50 x^{\frac{- 6 + 11}{11}}}{11} + 25 \frac{5}{11 x^{\frac{6}{11}}}$

Step 1.13.3.13.5

Add $- 6$ and $11$ .

$2 x^{\frac{16}{11}} - 10 x^{\frac{5}{11}} + \frac{5 x^{\frac{16}{11}}}{11} + \frac{- 50 x^{\frac{5}{11}}}{11} + 25 \frac{5}{11 x^{\frac{6}{11}}}$

Step 1.13.3.14

Move the negative in front of the fraction.

$2 x^{\frac{16}{11}} - 10 x^{\frac{5}{11}} + \frac{5 x^{\frac{16}{11}}}{11} - \frac{50 x^{\frac{5}{11}}}{11} + 25 \frac{5}{11 x^{\frac{6}{11}}}$

Step 1.13.3.15

Combine $25$ and $\frac{5}{11 x^{\frac{6}{11}}}$ .

$2 x^{\frac{16}{11}} - 10 x^{\frac{5}{11}} + \frac{5 x^{\frac{16}{11}}}{11} - \frac{50 x^{\frac{5}{11}}}{11} + \frac{25 \cdot 5}{11 x^{\frac{6}{11}}}$

Step 1.13.3.16

Multiply $25$ by $5$ .

$2 x^{\frac{16}{11}} - 10 x^{\frac{5}{11}} + \frac{5 x^{\frac{16}{11}}}{11} - \frac{50 x^{\frac{5}{11}}}{11} + \frac{125}{11 x^{\frac{6}{11}}}$

Step 1.13.3.17

To write $2 x^{\frac{16}{11}}$ as a fraction with a common denominator, multiply by $\frac{11}{11}$ .

$- 10 x^{\frac{5}{11}} + 2 x^{\frac{16}{11}} \cdot \frac{11}{11} + \frac{5 x^{\frac{16}{11}}}{11} - \frac{50 x^{\frac{5}{11}}}{11} + \frac{125}{11 x^{\frac{6}{11}}}$

Step 1.13.3.18

Combine $2 x^{\frac{16}{11}}$ and $\frac{11}{11}$ .

$- 10 x^{\frac{5}{11}} + \frac{2 x^{\frac{16}{11}} \cdot 11}{11} + \frac{5 x^{\frac{16}{11}}}{11} - \frac{50 x^{\frac{5}{11}}}{11} + \frac{125}{11 x^{\frac{6}{11}}}$

Step 1.13.3.19

Combine the numerators over the common denominator.

$- 10 x^{\frac{5}{11}} + \frac{2 x^{\frac{16}{11}} \cdot 11 + 5 x^{\frac{16}{11}}}{11} - \frac{50 x^{\frac{5}{11}}}{11} + \frac{125}{11 x^{\frac{6}{11}}}$

Step 1.13.3.20

Multiply $11$ by $2$ .

$- 10 x^{\frac{5}{11}} + \frac{22 x^{\frac{16}{11}} + 5 x^{\frac{16}{11}}}{11} - \frac{50 x^{\frac{5}{11}}}{11} + \frac{125}{11 x^{\frac{6}{11}}}$

Step 1.13.3.21

Add $22 x^{\frac{16}{11}}$ and $5 x^{\frac{16}{11}}$ .

$- 10 x^{\frac{5}{11}} + \frac{27 x^{\frac{16}{11}}}{11} - \frac{50 x^{\frac{5}{11}}}{11} + \frac{125}{11 x^{\frac{6}{11}}}$

Step 1.13.3.22

To write $- 10 x^{\frac{5}{11}}$ as a fraction with a common denominator, multiply by $\frac{11}{11}$ .

$- 10 x^{\frac{5}{11}} \cdot \frac{11}{11} - \frac{50 x^{\frac{5}{11}}}{11} + \frac{27 x^{\frac{16}{11}}}{11} + \frac{125}{11 x^{\frac{6}{11}}}$

Step 1.13.3.23

Combine $- 10 x^{\frac{5}{11}}$ and $\frac{11}{11}$ .

$\frac{- 10 x^{\frac{5}{11}} \cdot 11}{11} - \frac{50 x^{\frac{5}{11}}}{11} + \frac{27 x^{\frac{16}{11}}}{11} + \frac{125}{11 x^{\frac{6}{11}}}$

Step 1.13.3.24

Combine the numerators over the common denominator.

$\frac{- 10 x^{\frac{5}{11}} \cdot 11 - 50 x^{\frac{5}{11}}}{11} + \frac{27 x^{\frac{16}{11}}}{11} + \frac{125}{11 x^{\frac{6}{11}}}$

Step 1.13.3.25

Multiply $11$ by $- 10$ .

$\frac{- 110 x^{\frac{5}{11}} - 50 x^{\frac{5}{11}}}{11} + \frac{27 x^{\frac{16}{11}}}{11} + \frac{125}{11 x^{\frac{6}{11}}}$

Step 1.13.3.26

Subtract $50 x^{\frac{5}{11}}$ from $- 110 x^{\frac{5}{11}}$ .

$\frac{- 160 x^{\frac{5}{11}}}{11} + \frac{27 x^{\frac{16}{11}}}{11} + \frac{125}{11 x^{\frac{6}{11}}}$

Step 1.13.3.27

Move the negative in front of the fraction.

$- \frac{160 x^{\frac{5}{11}}}{11} + \frac{27 x^{\frac{16}{11}}}{11} + \frac{125}{11 x^{\frac{6}{11}}}$

Step 1.13.4

Reorder terms.

$\frac{27 x^{\frac{16}{11}}}{11} + \frac{125}{11 x^{\frac{6}{11}}} - \frac{160 x^{\frac{5}{11}}}{11}$

Step 2

Find the second derivative of the function.

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Step 2.1

By the Sum Rule, the derivative of $\frac{27 x^{\frac{16}{11}}}{11} + \frac{125}{11 x^{\frac{6}{11}}} - \frac{160 x^{\frac{5}{11}}}{11}$ with respect to $x$ is $\frac{d}{d x} [\frac{27 x^{\frac{16}{11}}}{11}] + \frac{d}{d x} [\frac{125}{11 x^{\frac{6}{11}}}] + \frac{d}{d x} [- \frac{160 x^{\frac{5}{11}}}{11}]$ .

$f'' (x) = \frac{d}{d x} (\frac{27 x^{\frac{16}{11}}}{11}) + \frac{d}{d x} (\frac{125}{11 x^{\frac{6}{11}}}) + \frac{d}{d x} (- \frac{160 x^{\frac{5}{11}}}{11})$

Step 2.2

Evaluate $\frac{d}{d x} [\frac{27 x^{\frac{16}{11}}}{11}]$ .

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Step 2.2.1

Since $\frac{27}{11}$ is constant with respect to $x$ , the derivative of $\frac{27 x^{\frac{16}{11}}}{11}$ with respect to $x$ is $\frac{27}{11} \frac{d}{d x} [x^{\frac{16}{11}}]$ .

$f'' (x) = \frac{27}{11} \cdot \frac{d}{d x} (x^{\frac{16}{11}}) + \frac{d}{d x} (\frac{125}{11 x^{\frac{6}{11}}}) + \frac{d}{d x} (- \frac{160 x^{\frac{5}{11}}}{11})$

Step 2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{16}{11}$ .

$f'' (x) = \frac{27}{11} \cdot (\frac{16}{11} x^{\frac{16}{11} - 1}) + \frac{d}{d x} (\frac{125}{11 x^{\frac{6}{11}}}) + \frac{d}{d x} (- \frac{160 x^{\frac{5}{11}}}{11})$

Step 2.2.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{11}{11}$ .

$f'' (x) = \frac{27}{11} \cdot (\frac{16}{11} x^{\frac{16}{11} - 1 \cdot \frac{11}{11}}) + \frac{d}{d x} (\frac{125}{11 x^{\frac{6}{11}}}) + \frac{d}{d x} (- \frac{160 x^{\frac{5}{11}}}{11})$

Step 2.2.4

Combine $- 1$ and $\frac{11}{11}$ .

$f'' (x) = \frac{27}{11} \cdot (\frac{16}{11} x^{\frac{16}{11} + \frac{- 1 \cdot 11}{11}}) + \frac{d}{d x} (\frac{125}{11 x^{\frac{6}{11}}}) + \frac{d}{d x} (- \frac{160 x^{\frac{5}{11}}}{11})$

Step 2.2.5

Combine the numerators over the common denominator.

$f'' (x) = \frac{27}{11} \cdot (\frac{16}{11} x^{\frac{16 - 1 \cdot 11}{11}}) + \frac{d}{d x} (\frac{125}{11 x^{\frac{6}{11}}}) + \frac{d}{d x} (- \frac{160 x^{\frac{5}{11}}}{11})$

Step 2.2.6

Simplify the numerator.

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Step 2.2.6.1

Multiply $- 1$ by $11$ .

$f'' (x) = \frac{27}{11} \cdot (\frac{16}{11} x^{\frac{16 - 11}{11}}) + \frac{d}{d x} (\frac{125}{11 x^{\frac{6}{11}}}) + \frac{d}{d x} (- \frac{160 x^{\frac{5}{11}}}{11})$

Step 2.2.6.2

Subtract $11$ from $16$ .

$f'' (x) = \frac{27}{11} \cdot (\frac{16}{11} x^{\frac{5}{11}}) + \frac{d}{d x} (\frac{125}{11 x^{\frac{6}{11}}}) + \frac{d}{d x} (- \frac{160 x^{\frac{5}{11}}}{11})$

Step 2.2.7

Combine $\frac{16}{11}$ and $x^{\frac{5}{11}}$ .

$f'' (x) = \frac{27}{11} \cdot \frac{16 x^{\frac{5}{11}}}{11} + \frac{d}{d x} (\frac{125}{11 x^{\frac{6}{11}}}) + \frac{d}{d x} (- \frac{160 x^{\frac{5}{11}}}{11})$

Step 2.2.8

Multiply $\frac{27}{11}$ by $\frac{16 x^{\frac{5}{11}}}{11}$ .

$f'' (x) = \frac{27 (16 x^{\frac{5}{11}})}{11 \cdot 11} + \frac{d}{d x} (\frac{125}{11 x^{\frac{6}{11}}}) + \frac{d}{d x} (- \frac{160 x^{\frac{5}{11}}}{11})$

Step 2.2.9

Multiply $16$ by $27$ .

$f'' (x) = \frac{432 x^{\frac{5}{11}}}{11 \cdot 11} + \frac{d}{d x} (\frac{125}{11 x^{\frac{6}{11}}}) + \frac{d}{d x} (- \frac{160 x^{\frac{5}{11}}}{11})$

Step 2.2.10

Multiply $11$ by $11$ .

$f'' (x) = \frac{432 x^{\frac{5}{11}}}{121} + \frac{d}{d x} (\frac{125}{11 x^{\frac{6}{11}}}) + \frac{d}{d x} (- \frac{160 x^{\frac{5}{11}}}{11})$

Step 2.3

Evaluate $\frac{d}{d x} [\frac{125}{11 x^{\frac{6}{11}}}]$ .

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Step 2.3.1

Since $\frac{125}{11}$ is constant with respect to $x$ , the derivative of $\frac{125}{11 x^{\frac{6}{11}}}$ with respect to $x$ is $\frac{125}{11} \frac{d}{d x} [\frac{1}{x^{\frac{6}{11}}}]$ .

$f'' (x) = \frac{432 x^{\frac{5}{11}}}{121} + \frac{125}{11} \cdot \frac{d}{d x} (\frac{1}{x^{\frac{6}{11}}}) + \frac{d}{d x} (- \frac{160 x^{\frac{5}{11}}}{11})$

Step 2.3.2

Rewrite $\frac{1}{x^{\frac{6}{11}}}$ as ${(x^{\frac{6}{11}})}^{- 1}$ .

$f'' (x) = \frac{432 x^{\frac{5}{11}}}{121} + \frac{125}{11} \cdot \frac{d}{d x} ({(x^{\frac{6}{11}})}^{- 1}) + \frac{d}{d x} (- \frac{160 x^{\frac{5}{11}}}{11})$

Step 2.3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{\frac{6}{11}}$ .

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Step 2.3.3.1

To apply the Chain Rule, set $u$ as $x^{\frac{6}{11}}$ .

$f'' (x) = \frac{432 x^{\frac{5}{11}}}{121} + \frac{125}{11} \cdot (\frac{d}{d u} (u^{- 1}) \frac{d}{d x} (x^{\frac{6}{11}})) + \frac{d}{d x} (- \frac{160 x^{\frac{5}{11}}}{11})$

Step 2.3.3.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = - 1$ .

$f'' (x) = \frac{432 x^{\frac{5}{11}}}{121} + \frac{125}{11} \cdot (- u^{- 2} \frac{d}{d x} x^{\frac{6}{11}}) + \frac{d}{d x} (- \frac{160 x^{\frac{5}{11}}}{11})$

Step 2.3.3.3

Replace all occurrences of $u$ with $x^{\frac{6}{11}}$ .

$f'' (x) = \frac{432 x^{\frac{5}{11}}}{121} + \frac{125}{11} \cdot (- {(x^{\frac{6}{11}})}^{- 2} \frac{d}{d x} x^{\frac{6}{11}}) + \frac{d}{d x} (- \frac{160 x^{\frac{5}{11}}}{11})$

Step 2.3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{6}{11}$ .

$f'' (x) = \frac{432 x^{\frac{5}{11}}}{121} + \frac{125}{11} \cdot (- {(x^{\frac{6}{11}})}^{- 2} (\frac{6}{11} x^{\frac{6}{11} - 1})) + \frac{d}{d x} (- \frac{160 x^{\frac{5}{11}}}{11})$

Step 2.3.5

Multiply the exponents in ${(x^{\frac{6}{11}})}^{- 2}$ .

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Step 2.3.5.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f'' (x) = \frac{432 x^{\frac{5}{11}}}{121} + \frac{125}{11} \cdot (- x^{\frac{6}{11} \cdot - 2} (\frac{6}{11} x^{\frac{6}{11} - 1})) + \frac{d}{d x} (- \frac{160 x^{\frac{5}{11}}}{11})$

Step 2.3.5.2

Multiply $\frac{6}{11} \cdot - 2$ .

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Step 2.3.5.2.1

Combine $\frac{6}{11}$ and $- 2$ .

$f'' (x) = \frac{432 x^{\frac{5}{11}}}{121} + \frac{125}{11} \cdot (- x^{\frac{6 \cdot - 2}{11}} (\frac{6}{11} x^{\frac{6}{11} - 1})) + \frac{d}{d x} (- \frac{160 x^{\frac{5}{11}}}{11})$

Step 2.3.5.2.2

Multiply $6$ by $- 2$ .

$f'' (x) = \frac{432 x^{\frac{5}{11}}}{121} + \frac{125}{11} \cdot (- x^{\frac{- 12}{11}} (\frac{6}{11} x^{\frac{6}{11} - 1})) + \frac{d}{d x} (- \frac{160 x^{\frac{5}{11}}}{11})$

Step 2.3.5.3

Move the negative in front of the fraction.

$f'' (x) = \frac{432 x^{\frac{5}{11}}}{121} + \frac{125}{11} \cdot (- x^{- \frac{12}{11}} (\frac{6}{11} x^{\frac{6}{11} - 1})) + \frac{d}{d x} (- \frac{160 x^{\frac{5}{11}}}{11})$

Step 2.3.6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{11}{11}$ .

$f'' (x) = \frac{432 x^{\frac{5}{11}}}{121} + \frac{125}{11} \cdot (- x^{- \frac{12}{11}} (\frac{6}{11} x^{\frac{6}{11} - 1 \cdot \frac{11}{11}})) + \frac{d}{d x} (- \frac{160 x^{\frac{5}{11}}}{11})$

Step 2.3.7

Combine $- 1$ and $\frac{11}{11}$ .

$f'' (x) = \frac{432 x^{\frac{5}{11}}}{121} + \frac{125}{11} \cdot (- x^{- \frac{12}{11}} (\frac{6}{11} x^{\frac{6}{11} + \frac{- 1 \cdot 11}{11}})) + \frac{d}{d x} (- \frac{160 x^{\frac{5}{11}}}{11})$

Step 2.3.8

Combine the numerators over the common denominator.

$f'' (x) = \frac{432 x^{\frac{5}{11}}}{121} + \frac{125}{11} \cdot (- x^{- \frac{12}{11}} (\frac{6}{11} x^{\frac{6 - 1 \cdot 11}{11}})) + \frac{d}{d x} (- \frac{160 x^{\frac{5}{11}}}{11})$

Step 2.3.9

Simplify the numerator.

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Step 2.3.9.1

Multiply $- 1$ by $11$ .

$f'' (x) = \frac{432 x^{\frac{5}{11}}}{121} + \frac{125}{11} \cdot (- x^{- \frac{12}{11}} (\frac{6}{11} x^{\frac{6 - 11}{11}})) + \frac{d}{d x} (- \frac{160 x^{\frac{5}{11}}}{11})$

Step 2.3.9.2

Subtract $11$ from $6$ .

$f'' (x) = \frac{432 x^{\frac{5}{11}}}{121} + \frac{125}{11} \cdot (- x^{- \frac{12}{11}} (\frac{6}{11} x^{\frac{- 5}{11}})) + \frac{d}{d x} (- \frac{160 x^{\frac{5}{11}}}{11})$

Step 2.3.10

Move the negative in front of the fraction.

$f'' (x) = \frac{432 x^{\frac{5}{11}}}{121} + \frac{125}{11} \cdot (- x^{- \frac{12}{11}} (\frac{6}{11} x^{- \frac{5}{11}})) + \frac{d}{d x} (- \frac{160 x^{\frac{5}{11}}}{11})$

Step 2.3.11

Combine $\frac{6}{11}$ and $x^{- \frac{5}{11}}$ .

$f'' (x) = \frac{432 x^{\frac{5}{11}}}{121} + \frac{125}{11} \cdot (- x^{- \frac{12}{11}} \frac{6 x^{- \frac{5}{11}}}{11}) + \frac{d}{d x} (- \frac{160 x^{\frac{5}{11}}}{11})$

Step 2.3.12

Combine $\frac{6 x^{- \frac{5}{11}}}{11}$ and $x^{- \frac{12}{11}}$ .

$f'' (x) = \frac{432 x^{\frac{5}{11}}}{121} + \frac{125}{11} \cdot (- \frac{6 x^{- \frac{5}{11}} x^{- \frac{12}{11}}}{11}) + \frac{d}{d x} (- \frac{160 x^{\frac{5}{11}}}{11})$

Step 2.3.13

Multiply $x^{- \frac{5}{11}}$ by $x^{- \frac{12}{11}}$ by adding the exponents.

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Step 2.3.13.1

Move $x^{- \frac{12}{11}}$ .

$f'' (x) = \frac{432 x^{\frac{5}{11}}}{121} + \frac{125}{11} \cdot (- \frac{6 (x^{- \frac{12}{11}} x^{- \frac{5}{11}})}{11}) + \frac{d}{d x} (- \frac{160 x^{\frac{5}{11}}}{11})$

Step 2.3.13.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = \frac{432 x^{\frac{5}{11}}}{121} + \frac{125}{11} \cdot (- \frac{6 x^{- \frac{12}{11} - \frac{5}{11}}}{11}) + \frac{d}{d x} (- \frac{160 x^{\frac{5}{11}}}{11})$

Step 2.3.13.3

Combine the numerators over the common denominator.

$f'' (x) = \frac{432 x^{\frac{5}{11}}}{121} + \frac{125}{11} \cdot (- \frac{6 x^{\frac{- 12 - 5}{11}}}{11}) + \frac{d}{d x} (- \frac{160 x^{\frac{5}{11}}}{11})$

Step 2.3.13.4

Subtract $5$ from $- 12$ .

$f'' (x) = \frac{432 x^{\frac{5}{11}}}{121} + \frac{125}{11} \cdot (- \frac{6 x^{\frac{- 17}{11}}}{11}) + \frac{d}{d x} (- \frac{160 x^{\frac{5}{11}}}{11})$

Step 2.3.13.5

Move the negative in front of the fraction.

$f'' (x) = \frac{432 x^{\frac{5}{11}}}{121} + \frac{125}{11} \cdot (- \frac{6 x^{- \frac{17}{11}}}{11}) + \frac{d}{d x} (- \frac{160 x^{\frac{5}{11}}}{11})$

Step 2.3.14

Move $x^{- \frac{17}{11}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (x) = \frac{432 x^{\frac{5}{11}}}{121} + \frac{125}{11} \cdot (- \frac{6}{11 x^{\frac{17}{11}}}) + \frac{d}{d x} (- \frac{160 x^{\frac{5}{11}}}{11})$

Step 2.3.15

Multiply $\frac{125}{11}$ by $\frac{6}{11 x^{\frac{17}{11}}}$ .

$f'' (x) = \frac{432 x^{\frac{5}{11}}}{121} - \frac{125 \cdot 6}{11 (11 x^{\frac{17}{11}})} + \frac{d}{d x} (- \frac{160 x^{\frac{5}{11}}}{11})$

Step 2.3.16

Multiply $125$ by $6$ .

$f'' (x) = \frac{432 x^{\frac{5}{11}}}{121} - \frac{750}{11 (11 x^{\frac{17}{11}})} + \frac{d}{d x} (- \frac{160 x^{\frac{5}{11}}}{11})$

Step 2.3.17

Multiply $11$ by $11$ .

$f'' (x) = \frac{432 x^{\frac{5}{11}}}{121} - \frac{750}{121 x^{\frac{17}{11}}} + \frac{d}{d x} (- \frac{160 x^{\frac{5}{11}}}{11})$

Step 2.4

Evaluate $\frac{d}{d x} [- \frac{160 x^{\frac{5}{11}}}{11}]$ .

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Step 2.4.1

Since $- \frac{160}{11}$ is constant with respect to $x$ , the derivative of $- \frac{160 x^{\frac{5}{11}}}{11}$ with respect to $x$ is $- \frac{160}{11} \frac{d}{d x} [x^{\frac{5}{11}}]$ .

$f'' (x) = \frac{432 x^{\frac{5}{11}}}{121} - \frac{750}{121 x^{\frac{17}{11}}} - \frac{160}{11} \cdot \frac{d}{d x} x^{\frac{5}{11}}$

Step 2.4.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{5}{11}$ .

$f'' (x) = \frac{432 x^{\frac{5}{11}}}{121} - \frac{750}{121 x^{\frac{17}{11}}} - \frac{160}{11} \cdot (\frac{5}{11} x^{\frac{5}{11} - 1})$

Step 2.4.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{11}{11}$ .

$f'' (x) = \frac{432 x^{\frac{5}{11}}}{121} - \frac{750}{121 x^{\frac{17}{11}}} - \frac{160}{11} \cdot (\frac{5}{11} x^{\frac{5}{11} - 1 \cdot \frac{11}{11}})$

Step 2.4.4

Combine $- 1$ and $\frac{11}{11}$ .

$f'' (x) = \frac{432 x^{\frac{5}{11}}}{121} - \frac{750}{121 x^{\frac{17}{11}}} - \frac{160}{11} \cdot (\frac{5}{11} x^{\frac{5}{11} + \frac{- 1 \cdot 11}{11}})$

Step 2.4.5

Combine the numerators over the common denominator.

$f'' (x) = \frac{432 x^{\frac{5}{11}}}{121} - \frac{750}{121 x^{\frac{17}{11}}} - \frac{160}{11} \cdot (\frac{5}{11} x^{\frac{5 - 1 \cdot 11}{11}})$

Step 2.4.6

Simplify the numerator.

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Step 2.4.6.1

Multiply $- 1$ by $11$ .

$f'' (x) = \frac{432 x^{\frac{5}{11}}}{121} - \frac{750}{121 x^{\frac{17}{11}}} - \frac{160}{11} \cdot (\frac{5}{11} x^{\frac{5 - 11}{11}})$

Step 2.4.6.2

Subtract $11$ from $5$ .

$f'' (x) = \frac{432 x^{\frac{5}{11}}}{121} - \frac{750}{121 x^{\frac{17}{11}}} - \frac{160}{11} \cdot (\frac{5}{11} x^{\frac{- 6}{11}})$

Step 2.4.7

Move the negative in front of the fraction.

$f'' (x) = \frac{432 x^{\frac{5}{11}}}{121} - \frac{750}{121 x^{\frac{17}{11}}} - \frac{160}{11} \cdot (\frac{5}{11} x^{- \frac{6}{11}})$

Step 2.4.8

Combine $\frac{5}{11}$ and $x^{- \frac{6}{11}}$ .

$f'' (x) = \frac{432 x^{\frac{5}{11}}}{121} - \frac{750}{121 x^{\frac{17}{11}}} - \frac{160}{11} \cdot \frac{5 x^{- \frac{6}{11}}}{11}$

Step 2.4.9

Multiply $\frac{5 x^{- \frac{6}{11}}}{11}$ by $\frac{160}{11}$ .

$f'' (x) = \frac{432 x^{\frac{5}{11}}}{121} - \frac{750}{121 x^{\frac{17}{11}}} - \frac{5 x^{- \frac{6}{11}} \cdot 160}{11 \cdot 11}$

Step 2.4.10

Multiply $160$ by $5$ .

$f'' (x) = \frac{432 x^{\frac{5}{11}}}{121} - \frac{750}{121 x^{\frac{17}{11}}} - \frac{800 x^{- \frac{6}{11}}}{11 \cdot 11}$

Step 2.4.11

Multiply $11$ by $11$ .

$f'' (x) = \frac{432 x^{\frac{5}{11}}}{121} - \frac{750}{121 x^{\frac{17}{11}}} - \frac{800 x^{- \frac{6}{11}}}{121}$

Step 2.4.12

Move $x^{- \frac{6}{11}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (x) = \frac{432 x^{\frac{5}{11}}}{121} - \frac{750}{121 x^{\frac{17}{11}}} - \frac{800}{121 x^{\frac{6}{11}}}$

Step 3

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$\frac{27 x^{\frac{16}{11}}}{11} + \frac{125}{11 x^{\frac{6}{11}}} - \frac{160 x^{\frac{5}{11}}}{11} = 0$

Step 4

Find the first derivative.

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Step 4.1

Find the first derivative.

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Step 4.1.1

Rewrite ${(x - 5)}^{2}$ as $(x - 5) (x - 5)$ .

$\frac{d}{d x} [x^{\frac{5}{11}} ((x - 5) (x - 5))]$

Step 4.1.2

Expand $(x - 5) (x - 5)$ using the FOIL Method.

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Step 4.1.2.1

Apply the distributive property.

$\frac{d}{d x} [x^{\frac{5}{11}} (x (x - 5) - 5 (x - 5))]$

Step 4.1.2.2

Apply the distributive property.

$\frac{d}{d x} [x^{\frac{5}{11}} (x \cdot x + x \cdot - 5 - 5 (x - 5))]$

Step 4.1.2.3

Apply the distributive property.

$\frac{d}{d x} [x^{\frac{5}{11}} (x \cdot x + x \cdot - 5 - 5 x - 5 \cdot - 5)]$

Step 4.1.3

Simplify and combine like terms.

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Step 4.1.3.1

Simplify each term.

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Step 4.1.3.1.1

Multiply $x$ by $x$ .

$\frac{d}{d x} [x^{\frac{5}{11}} (x^{2} + x \cdot - 5 - 5 x - 5 \cdot - 5)]$

Step 4.1.3.1.2

Move $- 5$ to the left of $x$ .

$\frac{d}{d x} [x^{\frac{5}{11}} (x^{2} - 5 \cdot x - 5 x - 5 \cdot - 5)]$

Step 4.1.3.1.3

Multiply $- 5$ by $- 5$ .

$\frac{d}{d x} [x^{\frac{5}{11}} (x^{2} - 5 x - 5 x + 25)]$

Step 4.1.3.2

Subtract $5 x$ from $- 5 x$ .

$\frac{d}{d x} [x^{\frac{5}{11}} (x^{2} - 10 x + 25)]$

Step 4.1.4

$x^{\frac{5}{11}} \frac{d}{d x} [x^{2} - 10 x + 25] + (x^{2} - 10 x + 25) \frac{d}{d x} [x^{\frac{5}{11}}]$

Step 4.1.5

Differentiate.

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Step 4.1.5.1

By the Sum Rule, the derivative of $x^{2} - 10 x + 25$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 10 x] + \frac{d}{d x} [25]$ .

$x^{\frac{5}{11}} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 10 x] + \frac{d}{d x} [25]) + (x^{2} - 10 x + 25) \frac{d}{d x} [x^{\frac{5}{11}}]$

Step 4.1.5.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$x^{\frac{5}{11}} (2 x + \frac{d}{d x} [- 10 x] + \frac{d}{d x} [25]) + (x^{2} - 10 x + 25) \frac{d}{d x} [x^{\frac{5}{11}}]$

Step 4.1.5.3

Since $- 10$ is constant with respect to $x$ , the derivative of $- 10 x$ with respect to $x$ is $- 10 \frac{d}{d x} [x]$ .

$x^{\frac{5}{11}} (2 x - 10 \frac{d}{d x} [x] + \frac{d}{d x} [25]) + (x^{2} - 10 x + 25) \frac{d}{d x} [x^{\frac{5}{11}}]$

Step 4.1.5.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$x^{\frac{5}{11}} (2 x - 10 \cdot 1 + \frac{d}{d x} [25]) + (x^{2} - 10 x + 25) \frac{d}{d x} [x^{\frac{5}{11}}]$

Step 4.1.5.5

Multiply $- 10$ by $1$ .

$x^{\frac{5}{11}} (2 x - 10 + \frac{d}{d x} [25]) + (x^{2} - 10 x + 25) \frac{d}{d x} [x^{\frac{5}{11}}]$

Step 4.1.5.6

Since $25$ is constant with respect to $x$ , the derivative of $25$ with respect to $x$ is $0$ .

$x^{\frac{5}{11}} (2 x - 10 + 0) + (x^{2} - 10 x + 25) \frac{d}{d x} [x^{\frac{5}{11}}]$

Step 4.1.5.7

Add $2 x - 10$ and $0$ .

$x^{\frac{5}{11}} (2 x - 10) + (x^{2} - 10 x + 25) \frac{d}{d x} [x^{\frac{5}{11}}]$

Step 4.1.5.8

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{5}{11}$ .

$x^{\frac{5}{11}} (2 x - 10) + (x^{2} - 10 x + 25) (\frac{5}{11} x^{\frac{5}{11} - 1})$

Step 4.1.6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{11}{11}$ .

$x^{\frac{5}{11}} (2 x - 10) + (x^{2} - 10 x + 25) (\frac{5}{11} x^{\frac{5}{11} - 1 \cdot \frac{11}{11}})$

Step 4.1.7

Combine $- 1$ and $\frac{11}{11}$ .

$x^{\frac{5}{11}} (2 x - 10) + (x^{2} - 10 x + 25) (\frac{5}{11} x^{\frac{5}{11} + \frac{- 1 \cdot 11}{11}})$

Step 4.1.8

Combine the numerators over the common denominator.

$x^{\frac{5}{11}} (2 x - 10) + (x^{2} - 10 x + 25) (\frac{5}{11} x^{\frac{5 - 1 \cdot 11}{11}})$

Step 4.1.9

Simplify the numerator.

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Step 4.1.9.1

Multiply $- 1$ by $11$ .

$x^{\frac{5}{11}} (2 x - 10) + (x^{2} - 10 x + 25) (\frac{5}{11} x^{\frac{5 - 11}{11}})$

Step 4.1.9.2

Subtract $11$ from $5$ .

$x^{\frac{5}{11}} (2 x - 10) + (x^{2} - 10 x + 25) (\frac{5}{11} x^{\frac{- 6}{11}})$

Step 4.1.10

Move the negative in front of the fraction.

$x^{\frac{5}{11}} (2 x - 10) + (x^{2} - 10 x + 25) (\frac{5}{11} x^{- \frac{6}{11}})$

Step 4.1.11

Combine $\frac{5}{11}$ and $x^{- \frac{6}{11}}$ .

$x^{\frac{5}{11}} (2 x - 10) + (x^{2} - 10 x + 25) \frac{5 x^{- \frac{6}{11}}}{11}$

Step 4.1.12

Move $x^{- \frac{6}{11}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$x^{\frac{5}{11}} (2 x - 10) + (x^{2} - 10 x + 25) \frac{5}{11 x^{\frac{6}{11}}}$

Step 4.1.13

Simplify.

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Step 4.1.13.1

Apply the distributive property.

$x^{\frac{5}{11}} (2 x) + x^{\frac{5}{11}} \cdot - 10 + (x^{2} - 10 x + 25) \frac{5}{11 x^{\frac{6}{11}}}$

Step 4.1.13.2

Apply the distributive property.

$x^{\frac{5}{11}} (2 x) + x^{\frac{5}{11}} \cdot - 10 + x^{2} \frac{5}{11 x^{\frac{6}{11}}} - 10 x \frac{5}{11 x^{\frac{6}{11}}} + 25 \frac{5}{11 x^{\frac{6}{11}}}$

Step 4.1.13.3

Combine terms.

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Step 4.1.13.3.1

Multiply $x^{\frac{5}{11}}$ by $x$ by adding the exponents.

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Step 4.1.13.3.1.1

Move $x$ .

$x \cdot x^{\frac{5}{11}} \cdot 2 + x^{\frac{5}{11}} \cdot - 10 + x^{2} \frac{5}{11 x^{\frac{6}{11}}} - 10 x \frac{5}{11 x^{\frac{6}{11}}} + 25 \frac{5}{11 x^{\frac{6}{11}}}$

Step 4.1.13.3.1.2

Multiply $x$ by $x^{\frac{5}{11}}$ .

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Step 4.1.13.3.1.2.1

Raise $x$ to the power of $1$ .

$x^{1} x^{\frac{5}{11}} \cdot 2 + x^{\frac{5}{11}} \cdot - 10 + x^{2} \frac{5}{11 x^{\frac{6}{11}}} - 10 x \frac{5}{11 x^{\frac{6}{11}}} + 25 \frac{5}{11 x^{\frac{6}{11}}}$

Step 4.1.13.3.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{1 + \frac{5}{11}} \cdot 2 + x^{\frac{5}{11}} \cdot - 10 + x^{2} \frac{5}{11 x^{\frac{6}{11}}} - 10 x \frac{5}{11 x^{\frac{6}{11}}} + 25 \frac{5}{11 x^{\frac{6}{11}}}$

Step 4.1.13.3.1.3

Write $1$ as a fraction with a common denominator.

$x^{\frac{11}{11} + \frac{5}{11}} \cdot 2 + x^{\frac{5}{11}} \cdot - 10 + x^{2} \frac{5}{11 x^{\frac{6}{11}}} - 10 x \frac{5}{11 x^{\frac{6}{11}}} + 25 \frac{5}{11 x^{\frac{6}{11}}}$

Step 4.1.13.3.1.4

Combine the numerators over the common denominator.

$x^{\frac{11 + 5}{11}} \cdot 2 + x^{\frac{5}{11}} \cdot - 10 + x^{2} \frac{5}{11 x^{\frac{6}{11}}} - 10 x \frac{5}{11 x^{\frac{6}{11}}} + 25 \frac{5}{11 x^{\frac{6}{11}}}$

Step 4.1.13.3.1.5

Add $11$ and $5$ .

$x^{\frac{16}{11}} \cdot 2 + x^{\frac{5}{11}} \cdot - 10 + x^{2} \frac{5}{11 x^{\frac{6}{11}}} - 10 x \frac{5}{11 x^{\frac{6}{11}}} + 25 \frac{5}{11 x^{\frac{6}{11}}}$

Step 4.1.13.3.2

Move $2$ to the left of $x^{\frac{16}{11}}$ .

$2 \cdot x^{\frac{16}{11}} + x^{\frac{5}{11}} \cdot - 10 + x^{2} \frac{5}{11 x^{\frac{6}{11}}} - 10 x \frac{5}{11 x^{\frac{6}{11}}} + 25 \frac{5}{11 x^{\frac{6}{11}}}$

Step 4.1.13.3.3

Move $- 10$ to the left of $x^{\frac{5}{11}}$ .

$2 x^{\frac{16}{11}} - 10 \cdot x^{\frac{5}{11}} + x^{2} \frac{5}{11 x^{\frac{6}{11}}} - 10 x \frac{5}{11 x^{\frac{6}{11}}} + 25 \frac{5}{11 x^{\frac{6}{11}}}$

Step 4.1.13.3.4

Combine $x^{2}$ and $\frac{5}{11 x^{\frac{6}{11}}}$ .

$2 x^{\frac{16}{11}} - 10 x^{\frac{5}{11}} + \frac{x^{2} \cdot 5}{11 x^{\frac{6}{11}}} - 10 x \frac{5}{11 x^{\frac{6}{11}}} + 25 \frac{5}{11 x^{\frac{6}{11}}}$

Step 4.1.13.3.5

Move $5$ to the left of $x^{2}$ .

$2 x^{\frac{16}{11}} - 10 x^{\frac{5}{11}} + \frac{5 \cdot x^{2}}{11 x^{\frac{6}{11}}} - 10 x \frac{5}{11 x^{\frac{6}{11}}} + 25 \frac{5}{11 x^{\frac{6}{11}}}$

Step 4.1.13.3.6

Move $x^{\frac{6}{11}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$2 x^{\frac{16}{11}} - 10 x^{\frac{5}{11}} + \frac{5 x^{2} x^{- \frac{6}{11}}}{11} - 10 x \frac{5}{11 x^{\frac{6}{11}}} + 25 \frac{5}{11 x^{\frac{6}{11}}}$

Step 4.1.13.3.7

Multiply $x^{2}$ by $x^{- \frac{6}{11}}$ by adding the exponents.

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Step 4.1.13.3.7.1

Move $x^{- \frac{6}{11}}$ .

$2 x^{\frac{16}{11}} - 10 x^{\frac{5}{11}} + \frac{5 (x^{- \frac{6}{11}} x^{2})}{11} - 10 x \frac{5}{11 x^{\frac{6}{11}}} + 25 \frac{5}{11 x^{\frac{6}{11}}}$

Step 4.1.13.3.7.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 x^{\frac{16}{11}} - 10 x^{\frac{5}{11}} + \frac{5 x^{- \frac{6}{11} + 2}}{11} - 10 x \frac{5}{11 x^{\frac{6}{11}}} + 25 \frac{5}{11 x^{\frac{6}{11}}}$

Step 4.1.13.3.7.3

To write $2$ as a fraction with a common denominator, multiply by $\frac{11}{11}$ .

$2 x^{\frac{16}{11}} - 10 x^{\frac{5}{11}} + \frac{5 x^{- \frac{6}{11} + 2 \cdot \frac{11}{11}}}{11} - 10 x \frac{5}{11 x^{\frac{6}{11}}} + 25 \frac{5}{11 x^{\frac{6}{11}}}$

Step 4.1.13.3.7.4

Combine $2$ and $\frac{11}{11}$ .

$2 x^{\frac{16}{11}} - 10 x^{\frac{5}{11}} + \frac{5 x^{- \frac{6}{11} + \frac{2 \cdot 11}{11}}}{11} - 10 x \frac{5}{11 x^{\frac{6}{11}}} + 25 \frac{5}{11 x^{\frac{6}{11}}}$

Step 4.1.13.3.7.5

Combine the numerators over the common denominator.

$2 x^{\frac{16}{11}} - 10 x^{\frac{5}{11}} + \frac{5 x^{\frac{- 6 + 2 \cdot 11}{11}}}{11} - 10 x \frac{5}{11 x^{\frac{6}{11}}} + 25 \frac{5}{11 x^{\frac{6}{11}}}$

Step 4.1.13.3.7.6

Simplify the numerator.

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Step 4.1.13.3.7.6.1

Multiply $2$ by $11$ .

$2 x^{\frac{16}{11}} - 10 x^{\frac{5}{11}} + \frac{5 x^{\frac{- 6 + 22}{11}}}{11} - 10 x \frac{5}{11 x^{\frac{6}{11}}} + 25 \frac{5}{11 x^{\frac{6}{11}}}$

Step 4.1.13.3.7.6.2

Add $- 6$ and $22$ .

$2 x^{\frac{16}{11}} - 10 x^{\frac{5}{11}} + \frac{5 x^{\frac{16}{11}}}{11} - 10 x \frac{5}{11 x^{\frac{6}{11}}} + 25 \frac{5}{11 x^{\frac{6}{11}}}$

Step 4.1.13.3.8

Combine $- 10$ and $\frac{5}{11 x^{\frac{6}{11}}}$ .

$2 x^{\frac{16}{11}} - 10 x^{\frac{5}{11}} + \frac{5 x^{\frac{16}{11}}}{11} + x \frac{- 10 \cdot 5}{11 x^{\frac{6}{11}}} + 25 \frac{5}{11 x^{\frac{6}{11}}}$

Step 4.1.13.3.9

Multiply $- 10$ by $5$ .

$2 x^{\frac{16}{11}} - 10 x^{\frac{5}{11}} + \frac{5 x^{\frac{16}{11}}}{11} + x \frac{- 50}{11 x^{\frac{6}{11}}} + 25 \frac{5}{11 x^{\frac{6}{11}}}$

Step 4.1.13.3.10

Combine $x$ and $\frac{- 50}{11 x^{\frac{6}{11}}}$ .

$2 x^{\frac{16}{11}} - 10 x^{\frac{5}{11}} + \frac{5 x^{\frac{16}{11}}}{11} + \frac{x \cdot - 50}{11 x^{\frac{6}{11}}} + 25 \frac{5}{11 x^{\frac{6}{11}}}$

Step 4.1.13.3.11

Move $- 50$ to the left of $x$ .

$2 x^{\frac{16}{11}} - 10 x^{\frac{5}{11}} + \frac{5 x^{\frac{16}{11}}}{11} + \frac{- 50 \cdot x}{11 x^{\frac{6}{11}}} + 25 \frac{5}{11 x^{\frac{6}{11}}}$

Step 4.1.13.3.12

Move $x^{\frac{6}{11}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$2 x^{\frac{16}{11}} - 10 x^{\frac{5}{11}} + \frac{5 x^{\frac{16}{11}}}{11} + \frac{- 50 x \cdot x^{- \frac{6}{11}}}{11} + 25 \frac{5}{11 x^{\frac{6}{11}}}$

Step 4.1.13.3.13

Multiply $x$ by $x^{- \frac{6}{11}}$ by adding the exponents.

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Step 4.1.13.3.13.1

Move $x^{- \frac{6}{11}}$ .

$2 x^{\frac{16}{11}} - 10 x^{\frac{5}{11}} + \frac{5 x^{\frac{16}{11}}}{11} + \frac{- 50 (x^{- \frac{6}{11}} x)}{11} + 25 \frac{5}{11 x^{\frac{6}{11}}}$

Step 4.1.13.3.13.2

Multiply $x^{- \frac{6}{11}}$ by $x$ .

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Step 4.1.13.3.13.2.1

Raise $x$ to the power of $1$ .

$2 x^{\frac{16}{11}} - 10 x^{\frac{5}{11}} + \frac{5 x^{\frac{16}{11}}}{11} + \frac{- 50 (x^{- \frac{6}{11}} x^{1})}{11} + 25 \frac{5}{11 x^{\frac{6}{11}}}$

Step 4.1.13.3.13.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 x^{\frac{16}{11}} - 10 x^{\frac{5}{11}} + \frac{5 x^{\frac{16}{11}}}{11} + \frac{- 50 x^{- \frac{6}{11} + 1}}{11} + 25 \frac{5}{11 x^{\frac{6}{11}}}$

Step 4.1.13.3.13.3

Write $1$ as a fraction with a common denominator.

$2 x^{\frac{16}{11}} - 10 x^{\frac{5}{11}} + \frac{5 x^{\frac{16}{11}}}{11} + \frac{- 50 x^{- \frac{6}{11} + \frac{11}{11}}}{11} + 25 \frac{5}{11 x^{\frac{6}{11}}}$

Step 4.1.13.3.13.4

Combine the numerators over the common denominator.

$2 x^{\frac{16}{11}} - 10 x^{\frac{5}{11}} + \frac{5 x^{\frac{16}{11}}}{11} + \frac{- 50 x^{\frac{- 6 + 11}{11}}}{11} + 25 \frac{5}{11 x^{\frac{6}{11}}}$

Step 4.1.13.3.13.5

Add $- 6$ and $11$ .

$2 x^{\frac{16}{11}} - 10 x^{\frac{5}{11}} + \frac{5 x^{\frac{16}{11}}}{11} + \frac{- 50 x^{\frac{5}{11}}}{11} + 25 \frac{5}{11 x^{\frac{6}{11}}}$

Step 4.1.13.3.14

Move the negative in front of the fraction.

$2 x^{\frac{16}{11}} - 10 x^{\frac{5}{11}} + \frac{5 x^{\frac{16}{11}}}{11} - \frac{50 x^{\frac{5}{11}}}{11} + 25 \frac{5}{11 x^{\frac{6}{11}}}$

Step 4.1.13.3.15

Combine $25$ and $\frac{5}{11 x^{\frac{6}{11}}}$ .

$2 x^{\frac{16}{11}} - 10 x^{\frac{5}{11}} + \frac{5 x^{\frac{16}{11}}}{11} - \frac{50 x^{\frac{5}{11}}}{11} + \frac{25 \cdot 5}{11 x^{\frac{6}{11}}}$

Step 4.1.13.3.16

Multiply $25$ by $5$ .

$2 x^{\frac{16}{11}} - 10 x^{\frac{5}{11}} + \frac{5 x^{\frac{16}{11}}}{11} - \frac{50 x^{\frac{5}{11}}}{11} + \frac{125}{11 x^{\frac{6}{11}}}$

Step 4.1.13.3.17

To write $2 x^{\frac{16}{11}}$ as a fraction with a common denominator, multiply by $\frac{11}{11}$ .

$- 10 x^{\frac{5}{11}} + 2 x^{\frac{16}{11}} \cdot \frac{11}{11} + \frac{5 x^{\frac{16}{11}}}{11} - \frac{50 x^{\frac{5}{11}}}{11} + \frac{125}{11 x^{\frac{6}{11}}}$

Step 4.1.13.3.18

Combine $2 x^{\frac{16}{11}}$ and $\frac{11}{11}$ .

$- 10 x^{\frac{5}{11}} + \frac{2 x^{\frac{16}{11}} \cdot 11}{11} + \frac{5 x^{\frac{16}{11}}}{11} - \frac{50 x^{\frac{5}{11}}}{11} + \frac{125}{11 x^{\frac{6}{11}}}$

Step 4.1.13.3.19

Combine the numerators over the common denominator.

$- 10 x^{\frac{5}{11}} + \frac{2 x^{\frac{16}{11}} \cdot 11 + 5 x^{\frac{16}{11}}}{11} - \frac{50 x^{\frac{5}{11}}}{11} + \frac{125}{11 x^{\frac{6}{11}}}$

Step 4.1.13.3.20

Multiply $11$ by $2$ .

$- 10 x^{\frac{5}{11}} + \frac{22 x^{\frac{16}{11}} + 5 x^{\frac{16}{11}}}{11} - \frac{50 x^{\frac{5}{11}}}{11} + \frac{125}{11 x^{\frac{6}{11}}}$

Step 4.1.13.3.21

Add $22 x^{\frac{16}{11}}$ and $5 x^{\frac{16}{11}}$ .

$- 10 x^{\frac{5}{11}} + \frac{27 x^{\frac{16}{11}}}{11} - \frac{50 x^{\frac{5}{11}}}{11} + \frac{125}{11 x^{\frac{6}{11}}}$

Step 4.1.13.3.22

To write $- 10 x^{\frac{5}{11}}$ as a fraction with a common denominator, multiply by $\frac{11}{11}$ .

$- 10 x^{\frac{5}{11}} \cdot \frac{11}{11} - \frac{50 x^{\frac{5}{11}}}{11} + \frac{27 x^{\frac{16}{11}}}{11} + \frac{125}{11 x^{\frac{6}{11}}}$

Step 4.1.13.3.23

Combine $- 10 x^{\frac{5}{11}}$ and $\frac{11}{11}$ .

$\frac{- 10 x^{\frac{5}{11}} \cdot 11}{11} - \frac{50 x^{\frac{5}{11}}}{11} + \frac{27 x^{\frac{16}{11}}}{11} + \frac{125}{11 x^{\frac{6}{11}}}$

Step 4.1.13.3.24

Combine the numerators over the common denominator.

$\frac{- 10 x^{\frac{5}{11}} \cdot 11 - 50 x^{\frac{5}{11}}}{11} + \frac{27 x^{\frac{16}{11}}}{11} + \frac{125}{11 x^{\frac{6}{11}}}$

Step 4.1.13.3.25

Multiply $11$ by $- 10$ .

$\frac{- 110 x^{\frac{5}{11}} - 50 x^{\frac{5}{11}}}{11} + \frac{27 x^{\frac{16}{11}}}{11} + \frac{125}{11 x^{\frac{6}{11}}}$

Step 4.1.13.3.26

Subtract $50 x^{\frac{5}{11}}$ from $- 110 x^{\frac{5}{11}}$ .

$\frac{- 160 x^{\frac{5}{11}}}{11} + \frac{27 x^{\frac{16}{11}}}{11} + \frac{125}{11 x^{\frac{6}{11}}}$

Step 4.1.13.3.27

Move the negative in front of the fraction.

$- \frac{160 x^{\frac{5}{11}}}{11} + \frac{27 x^{\frac{16}{11}}}{11} + \frac{125}{11 x^{\frac{6}{11}}}$

Step 4.1.13.4

Reorder terms.

$f' (x) = \frac{27 x^{\frac{16}{11}}}{11} + \frac{125}{11 x^{\frac{6}{11}}} - \frac{160 x^{\frac{5}{11}}}{11}$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{27 x^{\frac{16}{11}}}{11} + \frac{125}{11 x^{\frac{6}{11}}} - \frac{160 x^{\frac{5}{11}}}{11}$ .

$\frac{27 x^{\frac{16}{11}}}{11} + \frac{125}{11 x^{\frac{6}{11}}} - \frac{160 x^{\frac{5}{11}}}{11}$

Step 5

Set the first derivative equal to $0$ then solve the equation $\frac{27 x^{\frac{16}{11}}}{11} + \frac{125}{11 x^{\frac{6}{11}}} - \frac{160 x^{\frac{5}{11}}}{11} = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$\frac{27 x^{\frac{16}{11}}}{11} + \frac{125}{11 x^{\frac{6}{11}}} - \frac{160 x^{\frac{5}{11}}}{11} = 0$

Step 5.2

Find the LCD of the terms in the equation.

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Step 5.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$11, 11 x^{\frac{6}{11}}, 11, 1$

Step 5.2.2

Since $11, 11 x^{\frac{6}{11}}, 11, 1$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $11, 11, 11, 1$ then find LCM for the variable part $x^{\frac{6}{11}}$ .

Step 5.2.3

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 5.2.4

Since $11$ has no factors besides $1$ and $11$ .

$11$ is a prime number

Step 5.2.5

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 5.2.6

The LCM of $11, 11, 11, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$11$

Step 5.2.7

The LCM of $x^{\frac{6}{11}}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$x^{\frac{6}{11}}$

Step 5.2.8

The LCM for $11, 11 x^{\frac{6}{11}}, 11, 1$ is the numeric part $11$ multiplied by the variable part.

$11 x^{\frac{6}{11}}$

Step 5.3

Multiply each term in $\frac{27 x^{\frac{16}{11}}}{11} + \frac{125}{11 x^{\frac{6}{11}}} - \frac{160 x^{\frac{5}{11}}}{11} = 0$ by $11 x^{\frac{6}{11}}$ to eliminate the fractions.

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Step 5.3.1

Multiply each term in $\frac{27 x^{\frac{16}{11}}}{11} + \frac{125}{11 x^{\frac{6}{11}}} - \frac{160 x^{\frac{5}{11}}}{11} = 0$ by $11 x^{\frac{6}{11}}$ .

$\frac{27 x^{\frac{16}{11}}}{11} (11 x^{\frac{6}{11}}) + \frac{125}{11 x^{\frac{6}{11}}} (11 x^{\frac{6}{11}}) - \frac{160 x^{\frac{5}{11}}}{11} (11 x^{\frac{6}{11}}) = 0 (11 x^{\frac{6}{11}})$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Simplify each term.

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Step 5.3.2.1.1

Rewrite using the commutative property of multiplication.

$11 \frac{27 x^{\frac{16}{11}}}{11} x^{\frac{6}{11}} + \frac{125}{11 x^{\frac{6}{11}}} (11 x^{\frac{6}{11}}) - \frac{160 x^{\frac{5}{11}}}{11} (11 x^{\frac{6}{11}}) = 0 (11 x^{\frac{6}{11}})$

Step 5.3.2.1.2

Cancel the common factor of $11$ .

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Step 5.3.2.1.2.1

Cancel the common factor.

$11 \frac{27 x^{\frac{16}{11}}}{11} x^{\frac{6}{11}} + \frac{125}{11 x^{\frac{6}{11}}} (11 x^{\frac{6}{11}}) - \frac{160 x^{\frac{5}{11}}}{11} (11 x^{\frac{6}{11}}) = 0 (11 x^{\frac{6}{11}})$

Step 5.3.2.1.2.2

Rewrite the expression.

$27 x^{\frac{16}{11}} x^{\frac{6}{11}} + \frac{125}{11 x^{\frac{6}{11}}} (11 x^{\frac{6}{11}}) - \frac{160 x^{\frac{5}{11}}}{11} (11 x^{\frac{6}{11}}) = 0 (11 x^{\frac{6}{11}})$

Step 5.3.2.1.3

Multiply $x^{\frac{16}{11}}$ by $x^{\frac{6}{11}}$ by adding the exponents.

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Step 5.3.2.1.3.1

Move $x^{\frac{6}{11}}$ .

$27 (x^{\frac{6}{11}} x^{\frac{16}{11}}) + \frac{125}{11 x^{\frac{6}{11}}} (11 x^{\frac{6}{11}}) - \frac{160 x^{\frac{5}{11}}}{11} (11 x^{\frac{6}{11}}) = 0 (11 x^{\frac{6}{11}})$

Step 5.3.2.1.3.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$27 x^{\frac{6}{11} + \frac{16}{11}} + \frac{125}{11 x^{\frac{6}{11}}} (11 x^{\frac{6}{11}}) - \frac{160 x^{\frac{5}{11}}}{11} (11 x^{\frac{6}{11}}) = 0 (11 x^{\frac{6}{11}})$

Step 5.3.2.1.3.3

Combine the numerators over the common denominator.

$27 x^{\frac{6 + 16}{11}} + \frac{125}{11 x^{\frac{6}{11}}} (11 x^{\frac{6}{11}}) - \frac{160 x^{\frac{5}{11}}}{11} (11 x^{\frac{6}{11}}) = 0 (11 x^{\frac{6}{11}})$

Step 5.3.2.1.3.4

Add $6$ and $16$ .

$27 x^{\frac{22}{11}} + \frac{125}{11 x^{\frac{6}{11}}} (11 x^{\frac{6}{11}}) - \frac{160 x^{\frac{5}{11}}}{11} (11 x^{\frac{6}{11}}) = 0 (11 x^{\frac{6}{11}})$

Step 5.3.2.1.3.5

Divide $22$ by $11$ .

$27 x^{2} + \frac{125}{11 x^{\frac{6}{11}}} (11 x^{\frac{6}{11}}) - \frac{160 x^{\frac{5}{11}}}{11} (11 x^{\frac{6}{11}}) = 0 (11 x^{\frac{6}{11}})$

Step 5.3.2.1.4

Rewrite using the commutative property of multiplication.

$27 x^{2} + 11 \frac{125}{11 x^{\frac{6}{11}}} x^{\frac{6}{11}} - \frac{160 x^{\frac{5}{11}}}{11} (11 x^{\frac{6}{11}}) = 0 (11 x^{\frac{6}{11}})$

Step 5.3.2.1.5

Cancel the common factor of $11$ .

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Step 5.3.2.1.5.1

Cancel the common factor.

$27 x^{2} + 11 \frac{125}{11 x^{\frac{6}{11}}} x^{\frac{6}{11}} - \frac{160 x^{\frac{5}{11}}}{11} (11 x^{\frac{6}{11}}) = 0 (11 x^{\frac{6}{11}})$

Step 5.3.2.1.5.2

Rewrite the expression.

$27 x^{2} + \frac{125}{x^{\frac{6}{11}}} x^{\frac{6}{11}} - \frac{160 x^{\frac{5}{11}}}{11} (11 x^{\frac{6}{11}}) = 0 (11 x^{\frac{6}{11}})$

Step 5.3.2.1.6

Cancel the common factor of $x^{\frac{6}{11}}$ .

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Step 5.3.2.1.6.1

Cancel the common factor.

$27 x^{2} + \frac{125}{x^{\frac{6}{11}}} x^{\frac{6}{11}} - \frac{160 x^{\frac{5}{11}}}{11} (11 x^{\frac{6}{11}}) = 0 (11 x^{\frac{6}{11}})$

Step 5.3.2.1.6.2

Rewrite the expression.

$27 x^{2} + 125 - \frac{160 x^{\frac{5}{11}}}{11} (11 x^{\frac{6}{11}}) = 0 (11 x^{\frac{6}{11}})$

Step 5.3.2.1.7

Cancel the common factor of $11$ .

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Step 5.3.2.1.7.1

Move the leading negative in $- \frac{160 x^{\frac{5}{11}}}{11}$ into the numerator.

$27 x^{2} + 125 + \frac{- 160 x^{\frac{5}{11}}}{11} (11 x^{\frac{6}{11}}) = 0 (11 x^{\frac{6}{11}})$

Step 5.3.2.1.7.2

Factor $11$ out of $11 x^{\frac{6}{11}}$ .

$27 x^{2} + 125 + \frac{- 160 x^{\frac{5}{11}}}{11} (11 (x^{\frac{6}{11}})) = 0 (11 x^{\frac{6}{11}})$

Step 5.3.2.1.7.3

Cancel the common factor.

$27 x^{2} + 125 + \frac{- 160 x^{\frac{5}{11}}}{11} (11 x^{\frac{6}{11}}) = 0 (11 x^{\frac{6}{11}})$

Step 5.3.2.1.7.4

Rewrite the expression.

$27 x^{2} + 125 - 160 x^{\frac{5}{11}} x^{\frac{6}{11}} = 0 (11 x^{\frac{6}{11}})$

Step 5.3.2.1.8

Multiply $x^{\frac{5}{11}}$ by $x^{\frac{6}{11}}$ by adding the exponents.

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Step 5.3.2.1.8.1

Move $x^{\frac{6}{11}}$ .

$27 x^{2} + 125 - 160 (x^{\frac{6}{11}} x^{\frac{5}{11}}) = 0 (11 x^{\frac{6}{11}})$

Step 5.3.2.1.8.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$27 x^{2} + 125 - 160 x^{\frac{6}{11} + \frac{5}{11}} = 0 (11 x^{\frac{6}{11}})$

Step 5.3.2.1.8.3

Combine the numerators over the common denominator.

$27 x^{2} + 125 - 160 x^{\frac{6 + 5}{11}} = 0 (11 x^{\frac{6}{11}})$

Step 5.3.2.1.8.4

Add $6$ and $5$ .

$27 x^{2} + 125 - 160 x^{\frac{11}{11}} = 0 (11 x^{\frac{6}{11}})$

Step 5.3.2.1.8.5

Divide $11$ by $11$ .

$27 x^{2} + 125 - 160 x^{1} = 0 (11 x^{\frac{6}{11}})$

Step 5.3.2.1.9

Simplify $- 160 x^{1}$ .

$27 x^{2} + 125 - 160 x = 0 (11 x^{\frac{6}{11}})$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Multiply $0 (11 x^{\frac{6}{11}})$ .

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Step 5.3.3.1.1

Multiply $11$ by $0$ .

$27 x^{2} + 125 - 160 x = 0 x^{\frac{6}{11}}$

Step 5.3.3.1.2

Multiply $0$ by $x^{\frac{6}{11}}$ .

$27 x^{2} + 125 - 160 x = 0$

Step 5.4

Solve the equation.

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Step 5.4.1

Factor by grouping.

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Step 5.4.1.1

Reorder terms.

$27 x^{2} - 160 x + 125 = 0$

Step 5.4.1.2

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 27 \cdot 125 = 3375$ and whose sum is $b = - 160$ .

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Step 5.4.1.2.1

Factor $- 160$ out of $- 160 x$ .

$27 x^{2} - 160 x + 125 = 0$

Step 5.4.1.2.2

Rewrite $- 160$ as $- 25$ plus $- 135$

$27 x^{2} + (- 25 - 135) x + 125 = 0$

Step 5.4.1.2.3

Apply the distributive property.

$27 x^{2} - 25 x - 135 x + 125 = 0$

Step 5.4.1.3

Factor out the greatest common factor from each group.

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Step 5.4.1.3.1

Group the first two terms and the last two terms.

$(27 x^{2} - 25 x) - 135 x + 125 = 0$

Step 5.4.1.3.2

Factor out the greatest common factor (GCF) from each group.

$x (27 x - 25) - 5 (27 x - 25) = 0$

Step 5.4.1.4

Factor the polynomial by factoring out the greatest common factor, $27 x - 25$ .

$(27 x - 25) (x - 5) = 0$

Step 5.4.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$27 x - 25 = 0$

$x - 5 = 0$

Step 5.4.3

Set $27 x - 25$ equal to $0$ and solve for $x$ .

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Step 5.4.3.1

Set $27 x - 25$ equal to $0$ .

$27 x - 25 = 0$

Step 5.4.3.2

Solve $27 x - 25 = 0$ for $x$ .

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Step 5.4.3.2.1

Add $25$ to both sides of the equation.

$27 x = 25$

Step 5.4.3.2.2

Divide each term in $27 x = 25$ by $27$ and simplify.

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Step 5.4.3.2.2.1

Divide each term in $27 x = 25$ by $27$ .

$\frac{27 x}{27} = \frac{25}{27}$

Step 5.4.3.2.2.2

Simplify the left side.

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Step 5.4.3.2.2.2.1

Cancel the common factor of $27$ .

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Step 5.4.3.2.2.2.1.1

Cancel the common factor.

$\frac{27 x}{27} = \frac{25}{27}$

Step 5.4.3.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{25}{27}$

Step 5.4.4

Set $x - 5$ equal to $0$ and solve for $x$ .

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Step 5.4.4.1

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Step 5.4.4.2

Add $5$ to both sides of the equation.

$x = 5$

Step 5.4.5

The final solution is all the values that make $(27 x - 25) (x - 5) = 0$ true.

$x = \frac{25}{27}, 5$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

Convert expressions with fractional exponents to radicals.

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Step 6.1.1

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$\frac{27 \sqrt[11]{x^{16}}}{11} + \frac{125}{11 x^{\frac{6}{11}}} - \frac{160 x^{\frac{5}{11}}}{11}$

Step 6.1.2

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$\frac{27 \sqrt[11]{x^{16}}}{11} + \frac{125}{11 \sqrt[11]{x^{6}}} - \frac{160 x^{\frac{5}{11}}}{11}$

Step 6.1.3

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$\frac{27 \sqrt[11]{x^{16}}}{11} + \frac{125}{11 \sqrt[11]{x^{6}}} - \frac{160 \sqrt[11]{x^{5}}}{11}$

Step 6.2

Set the denominator in $\frac{125}{11 \sqrt[11]{x^{6}}}$ equal to $0$ to find where the expression is undefined.

$11 \sqrt[11]{x^{6}} = 0$

Step 6.3

Solve for $x$ .

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Step 6.3.1

To remove the radical on the left side of the equation, raise both sides of the equation to the power of $11$ .

${(11 \sqrt[11]{x^{6}})}^{11} = 0^{11}$

Step 6.3.2

Simplify each side of the equation.

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Step 6.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[11]{x^{6}}$ as $x^{\frac{6}{11}}$ .

${(11 x^{\frac{6}{11}})}^{11} = 0^{11}$

Step 6.3.2.2

Simplify the left side.

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Step 6.3.2.2.1

Simplify ${(11 x^{\frac{6}{11}})}^{11}$ .

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Step 6.3.2.2.1.1

Apply the product rule to $11 x^{\frac{6}{11}}$ .

$11^{11} {(x^{\frac{6}{11}})}^{11} = 0^{11}$

Step 6.3.2.2.1.2

Raise $11$ to the power of $11$ .

$285311670611 {(x^{\frac{6}{11}})}^{11} = 0^{11}$

Step 6.3.2.2.1.3

Multiply the exponents in ${(x^{\frac{6}{11}})}^{11}$ .

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Step 6.3.2.2.1.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$285311670611 x^{\frac{6}{11} \cdot 11} = 0^{11}$

Step 6.3.2.2.1.3.2

Cancel the common factor of $11$ .

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Step 6.3.2.2.1.3.2.1

Cancel the common factor.

$285311670611 x^{\frac{6}{11} \cdot 11} = 0^{11}$

Step 6.3.2.2.1.3.2.2

Rewrite the expression.

$285311670611 x^{6} = 0^{11}$

Step 6.3.2.3

Simplify the right side.

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Step 6.3.2.3.1

Raising $0$ to any positive power yields $0$ .

$285311670611 x^{6} = 0$

Step 6.3.3

Solve for $x$ .

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Step 6.3.3.1

Divide each term in $285311670611 x^{6} = 0$ by $285311670611$ and simplify.

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Step 6.3.3.1.1

Divide each term in $285311670611 x^{6} = 0$ by $285311670611$ .

$\frac{285311670611 x^{6}}{285311670611} = \frac{0}{285311670611}$

Step 6.3.3.1.2

Simplify the left side.

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Step 6.3.3.1.2.1

Cancel the common factor of $285311670611$ .

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Step 6.3.3.1.2.1.1

Cancel the common factor.

$\frac{285311670611 x^{6}}{285311670611} = \frac{0}{285311670611}$

Step 6.3.3.1.2.1.2

Divide $x^{6}$ by $1$ .

$x^{6} = \frac{0}{285311670611}$

Step 6.3.3.1.3

Simplify the right side.

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Step 6.3.3.1.3.1

Divide $0$ by $285311670611$ .

$x^{6} = 0$

Step 6.3.3.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt[6]{0}$

Step 6.3.3.3

Simplify $\pm \sqrt[6]{0}$ .

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Step 6.3.3.3.1

Rewrite $0$ as $0^{6}$ .

$x = \pm \sqrt[6]{0^{6}}$

Step 6.3.3.3.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Step 6.3.3.3.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 7

Critical points to evaluate.

$x = \frac{25}{27}, 5, 0$

Step 8

Evaluate the second derivative at $x = \frac{25}{27}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{432 {(\frac{25}{27})}^{\frac{5}{11}}}{121} - \frac{750}{121 {(\frac{25}{27})}^{\frac{17}{11}}} - \frac{800}{121 {(\frac{25}{27})}^{\frac{6}{11}}}$

Step 9

Evaluate the second derivative.

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Step 9.1

Simplify each term.

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Step 9.1.1

Apply the product rule to $\frac{25}{27}$ .

$\frac{432 \frac{25^{\frac{5}{11}}}{27^{\frac{5}{11}}}}{121} - \frac{750}{121 {(\frac{25}{27})}^{\frac{17}{11}}} - \frac{800}{121 {(\frac{25}{27})}^{\frac{6}{11}}}$

Step 9.1.2

Combine $432$ and $\frac{25^{\frac{5}{11}}}{27^{\frac{5}{11}}}$ .

$\frac{\frac{432 \cdot 25^{\frac{5}{11}}}{27^{\frac{5}{11}}}}{121} - \frac{750}{121 {(\frac{25}{27})}^{\frac{17}{11}}} - \frac{800}{121 {(\frac{25}{27})}^{\frac{6}{11}}}$

Step 9.1.3

Multiply the numerator by the reciprocal of the denominator.

$\frac{432 \cdot 25^{\frac{5}{11}}}{27^{\frac{5}{11}}} \cdot \frac{1}{121} - \frac{750}{121 {(\frac{25}{27})}^{\frac{17}{11}}} - \frac{800}{121 {(\frac{25}{27})}^{\frac{6}{11}}}$

Step 9.1.4

Combine.

$\frac{432 \cdot 25^{\frac{5}{11}} \cdot 1}{27^{\frac{5}{11}} \cdot 121} - \frac{750}{121 {(\frac{25}{27})}^{\frac{17}{11}}} - \frac{800}{121 {(\frac{25}{27})}^{\frac{6}{11}}}$

Step 9.1.5

Multiply $432$ by $1$ .

$\frac{432 \cdot 25^{\frac{5}{11}}}{27^{\frac{5}{11}} \cdot 121} - \frac{750}{121 {(\frac{25}{27})}^{\frac{17}{11}}} - \frac{800}{121 {(\frac{25}{27})}^{\frac{6}{11}}}$

Step 9.1.6

Move $121$ to the left of $27^{\frac{5}{11}}$ .

$\frac{432 \cdot 25^{\frac{5}{11}}}{121 \cdot 27^{\frac{5}{11}}} - \frac{750}{121 {(\frac{25}{27})}^{\frac{17}{11}}} - \frac{800}{121 {(\frac{25}{27})}^{\frac{6}{11}}}$

Step 9.1.7

Apply the product rule to $\frac{25}{27}$ .

$\frac{432 \cdot 25^{\frac{5}{11}}}{121 \cdot 27^{\frac{5}{11}}} - \frac{750}{121 \frac{25^{\frac{17}{11}}}{27^{\frac{17}{11}}}} - \frac{800}{121 {(\frac{25}{27})}^{\frac{6}{11}}}$

Step 9.1.8

Combine $121$ and $\frac{25^{\frac{17}{11}}}{27^{\frac{17}{11}}}$ .

$\frac{432 \cdot 25^{\frac{5}{11}}}{121 \cdot 27^{\frac{5}{11}}} - \frac{750}{\frac{121 \cdot 25^{\frac{17}{11}}}{27^{\frac{17}{11}}}} - \frac{800}{121 {(\frac{25}{27})}^{\frac{6}{11}}}$

Step 9.1.9

Multiply the numerator by the reciprocal of the denominator.

$\frac{432 \cdot 25^{\frac{5}{11}}}{121 \cdot 27^{\frac{5}{11}}} - (750 \frac{27^{\frac{17}{11}}}{121 \cdot 25^{\frac{17}{11}}}) - \frac{800}{121 {(\frac{25}{27})}^{\frac{6}{11}}}$

Step 9.1.10

Combine $750$ and $\frac{27^{\frac{17}{11}}}{121 \cdot 25^{\frac{17}{11}}}$ .

$\frac{432 \cdot 25^{\frac{5}{11}}}{121 \cdot 27^{\frac{5}{11}}} - \frac{750 \cdot 27^{\frac{17}{11}}}{121 \cdot 25^{\frac{17}{11}}} - \frac{800}{121 {(\frac{25}{27})}^{\frac{6}{11}}}$

Step 9.1.11

Apply the product rule to $\frac{25}{27}$ .

$\frac{432 \cdot 25^{\frac{5}{11}}}{121 \cdot 27^{\frac{5}{11}}} - \frac{750 \cdot 27^{\frac{17}{11}}}{121 \cdot 25^{\frac{17}{11}}} - \frac{800}{121 \frac{25^{\frac{6}{11}}}{27^{\frac{6}{11}}}}$

Step 9.1.12

Combine $121$ and $\frac{25^{\frac{6}{11}}}{27^{\frac{6}{11}}}$ .

$\frac{432 \cdot 25^{\frac{5}{11}}}{121 \cdot 27^{\frac{5}{11}}} - \frac{750 \cdot 27^{\frac{17}{11}}}{121 \cdot 25^{\frac{17}{11}}} - \frac{800}{\frac{121 \cdot 25^{\frac{6}{11}}}{27^{\frac{6}{11}}}}$

Step 9.1.13

Multiply the numerator by the reciprocal of the denominator.

$\frac{432 \cdot 25^{\frac{5}{11}}}{121 \cdot 27^{\frac{5}{11}}} - \frac{750 \cdot 27^{\frac{17}{11}}}{121 \cdot 25^{\frac{17}{11}}} - (800 \frac{27^{\frac{6}{11}}}{121 \cdot 25^{\frac{6}{11}}})$

Step 9.1.14

Combine $800$ and $\frac{27^{\frac{6}{11}}}{121 \cdot 25^{\frac{6}{11}}}$ .

$\frac{432 \cdot 25^{\frac{5}{11}}}{121 \cdot 27^{\frac{5}{11}}} - \frac{750 \cdot 27^{\frac{17}{11}}}{121 \cdot 25^{\frac{17}{11}}} - \frac{800 \cdot 27^{\frac{6}{11}}}{121 \cdot 25^{\frac{6}{11}}}$

Step 9.2

To write $- \frac{800 \cdot 27^{\frac{6}{11}}}{121 \cdot 25^{\frac{6}{11}}}$ as a fraction with a common denominator, multiply by $\frac{25^{\frac{11}{11}}}{25^{\frac{11}{11}}}$ .

$\frac{432 \cdot 25^{\frac{5}{11}}}{121 \cdot 27^{\frac{5}{11}}} - \frac{750 \cdot 27^{\frac{17}{11}}}{121 \cdot 25^{\frac{17}{11}}} - \frac{800 \cdot 27^{\frac{6}{11}}}{121 \cdot 25^{\frac{6}{11}}} \cdot \frac{25^{\frac{11}{11}}}{25^{\frac{11}{11}}}$

Step 9.3

Write each expression with a common denominator of $121 \cdot 25^{\frac{17}{11}}$ , by multiplying each by an appropriate factor of $1$ .

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Step 9.3.1

Multiply $\frac{800 \cdot 27^{\frac{6}{11}}}{121 \cdot 25^{\frac{6}{11}}}$ by $\frac{25^{\frac{11}{11}}}{25^{\frac{11}{11}}}$ .

$\frac{432 \cdot 25^{\frac{5}{11}}}{121 \cdot 27^{\frac{5}{11}}} - \frac{750 \cdot 27^{\frac{17}{11}}}{121 \cdot 25^{\frac{17}{11}}} - \frac{800 \cdot 27^{\frac{6}{11}} \cdot 25^{\frac{11}{11}}}{121 \cdot 25^{\frac{6}{11}} \cdot 25^{\frac{11}{11}}}$

Step 9.3.2

Multiply $25^{\frac{6}{11}}$ by $25^{\frac{11}{11}}$ by adding the exponents.

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Step 9.3.2.1

Move $25^{\frac{11}{11}}$ .

$\frac{432 \cdot 25^{\frac{5}{11}}}{121 \cdot 27^{\frac{5}{11}}} - \frac{750 \cdot 27^{\frac{17}{11}}}{121 \cdot 25^{\frac{17}{11}}} - \frac{800 \cdot 27^{\frac{6}{11}} \cdot 25^{\frac{11}{11}}}{121 \cdot (25^{\frac{11}{11}} \cdot 25^{\frac{6}{11}})}$

Step 9.3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{432 \cdot 25^{\frac{5}{11}}}{121 \cdot 27^{\frac{5}{11}}} - \frac{750 \cdot 27^{\frac{17}{11}}}{121 \cdot 25^{\frac{17}{11}}} - \frac{800 \cdot 27^{\frac{6}{11}} \cdot 25^{\frac{11}{11}}}{121 \cdot 25^{\frac{11}{11} + \frac{6}{11}}}$

Step 9.3.2.3

Combine the numerators over the common denominator.

$\frac{432 \cdot 25^{\frac{5}{11}}}{121 \cdot 27^{\frac{5}{11}}} - \frac{750 \cdot 27^{\frac{17}{11}}}{121 \cdot 25^{\frac{17}{11}}} - \frac{800 \cdot 27^{\frac{6}{11}} \cdot 25^{\frac{11}{11}}}{121 \cdot 25^{\frac{11 + 6}{11}}}$

Step 9.3.2.4

Add $11$ and $6$ .

$\frac{432 \cdot 25^{\frac{5}{11}}}{121 \cdot 27^{\frac{5}{11}}} - \frac{750 \cdot 27^{\frac{17}{11}}}{121 \cdot 25^{\frac{17}{11}}} - \frac{800 \cdot 27^{\frac{6}{11}} \cdot 25^{\frac{11}{11}}}{121 \cdot 25^{\frac{17}{11}}}$

Step 9.4

Combine the numerators over the common denominator.

$\frac{432 \cdot 25^{\frac{5}{11}}}{121 \cdot 27^{\frac{5}{11}}} + \frac{- 750 \cdot 27^{\frac{17}{11}} - 800 \cdot 27^{\frac{6}{11}} \cdot 25^{\frac{11}{11}}}{121 \cdot 25^{\frac{17}{11}}}$

Step 9.5

Simplify each term.

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Step 9.5.1

Cancel the common factor of $11$ .

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Step 9.5.1.1

Cancel the common factor.

$\frac{432 \cdot 25^{\frac{5}{11}}}{121 \cdot 27^{\frac{5}{11}}} + \frac{- 750 \cdot 27^{\frac{17}{11}} - 800 \cdot 27^{\frac{6}{11}} \cdot 25^{\frac{11}{11}}}{121 \cdot 25^{\frac{17}{11}}}$

Step 9.5.1.2

Rewrite the expression.

$\frac{432 \cdot 25^{\frac{5}{11}}}{121 \cdot 27^{\frac{5}{11}}} + \frac{- 750 \cdot 27^{\frac{17}{11}} - 800 \cdot 27^{\frac{6}{11}} \cdot 25^{1}}{121 \cdot 25^{\frac{17}{11}}}$

Step 9.5.2

Simplify the numerator.

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Step 9.5.2.1

Evaluate the exponent.

$\frac{432 \cdot 25^{\frac{5}{11}}}{121 \cdot 27^{\frac{5}{11}}} + \frac{- 750 \cdot 27^{\frac{17}{11}} - 800 \cdot 27^{\frac{6}{11}} \cdot 25}{121 \cdot 25^{\frac{17}{11}}}$

Step 9.5.2.2

Multiply $25$ by $- 800$ .

$\frac{432 \cdot 25^{\frac{5}{11}}}{121 \cdot 27^{\frac{5}{11}}} + \frac{- 750 \cdot 27^{\frac{17}{11}} - 20000 \cdot 27^{\frac{6}{11}}}{121 \cdot 25^{\frac{17}{11}}}$

Step 10

$x = \frac{25}{27}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = \frac{25}{27}$ is a local maximum

Step 11

Find the y-value when $x = \frac{25}{27}$ .

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Step 11.1

Replace the variable $x$ with $\frac{25}{27}$ in the expression.

$f (\frac{25}{27}) = {(\frac{25}{27})}^{\frac{5}{11}} {((\frac{25}{27}) - 5)}^{2}$

Step 11.2

Simplify the result.

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Step 11.2.1

Apply the product rule to $\frac{25}{27}$ .

$f (\frac{25}{27}) = \frac{25^{\frac{5}{11}}}{27^{\frac{5}{11}}} \cdot {((\frac{25}{27}) - 5)}^{2}$

Step 11.2.2

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{27}{27}$ .

$f (\frac{25}{27}) = \frac{25^{\frac{5}{11}}}{27^{\frac{5}{11}}} \cdot {(\frac{25}{27} - 5 \cdot \frac{27}{27})}^{2}$

Step 11.2.3

Combine $- 5$ and $\frac{27}{27}$ .

$f (\frac{25}{27}) = \frac{25^{\frac{5}{11}}}{27^{\frac{5}{11}}} \cdot {(\frac{25}{27} + \frac{- 5 \cdot 27}{27})}^{2}$

Step 11.2.4

Combine the numerators over the common denominator.

$f (\frac{25}{27}) = \frac{25^{\frac{5}{11}}}{27^{\frac{5}{11}}} \cdot {(\frac{25 - 5 \cdot 27}{27})}^{2}$

Step 11.2.5

Simplify the numerator.

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Step 11.2.5.1

Multiply $- 5$ by $27$ .

$f (\frac{25}{27}) = \frac{25^{\frac{5}{11}}}{27^{\frac{5}{11}}} \cdot {(\frac{25 - 135}{27})}^{2}$

Step 11.2.5.2

Subtract $135$ from $25$ .

$f (\frac{25}{27}) = \frac{25^{\frac{5}{11}}}{27^{\frac{5}{11}}} \cdot {(\frac{- 110}{27})}^{2}$

Step 11.2.6

Move the negative in front of the fraction.

$f (\frac{25}{27}) = \frac{25^{\frac{5}{11}}}{27^{\frac{5}{11}}} \cdot {(- \frac{110}{27})}^{2}$

Step 11.2.7

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 11.2.7.1

Apply the product rule to $- \frac{110}{27}$ .

$f (\frac{25}{27}) = \frac{25^{\frac{5}{11}}}{27^{\frac{5}{11}}} \cdot ({(- 1)}^{2} {(\frac{110}{27})}^{2})$

Step 11.2.7.2

Apply the product rule to $\frac{110}{27}$ .

$f (\frac{25}{27}) = \frac{25^{\frac{5}{11}}}{27^{\frac{5}{11}}} \cdot ({(- 1)}^{2} (\frac{110^{2}}{27^{2}}))$

Step 11.2.8

Simplify the expression.

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Step 11.2.8.1

Raise $- 1$ to the power of $2$ .

$f (\frac{25}{27}) = \frac{25^{\frac{5}{11}}}{27^{\frac{5}{11}}} \cdot (1 (\frac{110^{2}}{27^{2}}))$

Step 11.2.8.2

Multiply $\frac{110^{2}}{27^{2}}$ by $1$ .

$f (\frac{25}{27}) = \frac{25^{\frac{5}{11}}}{27^{\frac{5}{11}}} \cdot \frac{110^{2}}{27^{2}}$

Step 11.2.9

Combine.

$f (\frac{25}{27}) = \frac{25^{\frac{5}{11}} \cdot 110^{2}}{27^{\frac{5}{11}} \cdot 27^{2}}$

Step 11.2.10

Multiply $27^{\frac{5}{11}}$ by $27^{2}$ by adding the exponents.

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Step 11.2.10.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (\frac{25}{27}) = \frac{25^{\frac{5}{11}} \cdot 110^{2}}{27^{\frac{5}{11} + 2}}$

Step 11.2.10.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{11}{11}$ .

$f (\frac{25}{27}) = \frac{25^{\frac{5}{11}} \cdot 110^{2}}{27^{\frac{5}{11} + 2 \cdot \frac{11}{11}}}$

Step 11.2.10.3

Combine $2$ and $\frac{11}{11}$ .

$f (\frac{25}{27}) = \frac{25^{\frac{5}{11}} \cdot 110^{2}}{27^{\frac{5}{11} + \frac{2 \cdot 11}{11}}}$

Step 11.2.10.4

Combine the numerators over the common denominator.

$f (\frac{25}{27}) = \frac{25^{\frac{5}{11}} \cdot 110^{2}}{27^{\frac{5 + 2 \cdot 11}{11}}}$

Step 11.2.10.5

Simplify the numerator.

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Step 11.2.10.5.1

Multiply $2$ by $11$ .

$f (\frac{25}{27}) = \frac{25^{\frac{5}{11}} \cdot 110^{2}}{27^{\frac{5 + 22}{11}}}$

Step 11.2.10.5.2

Add $5$ and $22$ .

$f (\frac{25}{27}) = \frac{25^{\frac{5}{11}} \cdot 110^{2}}{27^{\frac{27}{11}}}$

Step 11.2.11

Raise $110$ to the power of $2$ .

$f (\frac{25}{27}) = \frac{25^{\frac{5}{11}} \cdot 12100}{27^{\frac{27}{11}}}$

Step 11.2.12

Move $12100$ to the left of $25^{\frac{5}{11}}$ .

$f (\frac{25}{27}) = \frac{12100 \cdot 25^{\frac{5}{11}}}{27^{\frac{27}{11}}}$

Step 11.2.13

The final answer is $\frac{12100 \cdot 25^{\frac{5}{11}}}{27^{\frac{27}{11}}}$ .

$y = \frac{12100 \cdot 25^{\frac{5}{11}}}{27^{\frac{27}{11}}}$

Step 12

Evaluate the second derivative at $x = 5$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{432 {(5)}^{\frac{5}{11}}}{121} - \frac{750}{121 {(5)}^{\frac{17}{11}}} - \frac{800}{121 {(5)}^{\frac{6}{11}}}$

Step 13

Evaluate the second derivative.

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Step 13.1

Remove parentheses.

$\frac{432 \cdot 5^{\frac{5}{11}}}{121} - \frac{750}{121 \cdot 5^{\frac{17}{11}}} - \frac{800}{121 \cdot 5^{\frac{6}{11}}}$

Step 13.2

To write $- \frac{800}{121 \cdot 5^{\frac{6}{11}}}$ as a fraction with a common denominator, multiply by $\frac{5^{\frac{11}{11}}}{5^{\frac{11}{11}}}$ .

$\frac{432 \cdot 5^{\frac{5}{11}}}{121} - \frac{750}{121 \cdot 5^{\frac{17}{11}}} - \frac{800}{121 \cdot 5^{\frac{6}{11}}} \cdot \frac{5^{\frac{11}{11}}}{5^{\frac{11}{11}}}$

Step 13.3

Write each expression with a common denominator of $121 \cdot 5^{\frac{17}{11}}$ , by multiplying each by an appropriate factor of $1$ .

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Step 13.3.1

Multiply $\frac{800}{121 \cdot 5^{\frac{6}{11}}}$ by $\frac{5^{\frac{11}{11}}}{5^{\frac{11}{11}}}$ .

$\frac{432 \cdot 5^{\frac{5}{11}}}{121} - \frac{750}{121 \cdot 5^{\frac{17}{11}}} - \frac{800 \cdot 5^{\frac{11}{11}}}{121 \cdot 5^{\frac{6}{11}} \cdot 5^{\frac{11}{11}}}$

Step 13.3.2

Multiply $5^{\frac{6}{11}}$ by $5^{\frac{11}{11}}$ by adding the exponents.

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Step 13.3.2.1

Move $5^{\frac{11}{11}}$ .

$\frac{432 \cdot 5^{\frac{5}{11}}}{121} - \frac{750}{121 \cdot 5^{\frac{17}{11}}} - \frac{800 \cdot 5^{\frac{11}{11}}}{121 \cdot (5^{\frac{11}{11}} \cdot 5^{\frac{6}{11}})}$

Step 13.3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{432 \cdot 5^{\frac{5}{11}}}{121} - \frac{750}{121 \cdot 5^{\frac{17}{11}}} - \frac{800 \cdot 5^{\frac{11}{11}}}{121 \cdot 5^{\frac{11}{11} + \frac{6}{11}}}$

Step 13.3.2.3

Combine the numerators over the common denominator.

$\frac{432 \cdot 5^{\frac{5}{11}}}{121} - \frac{750}{121 \cdot 5^{\frac{17}{11}}} - \frac{800 \cdot 5^{\frac{11}{11}}}{121 \cdot 5^{\frac{11 + 6}{11}}}$

Step 13.3.2.4

Add $11$ and $6$ .

$\frac{432 \cdot 5^{\frac{5}{11}}}{121} - \frac{750}{121 \cdot 5^{\frac{17}{11}}} - \frac{800 \cdot 5^{\frac{11}{11}}}{121 \cdot 5^{\frac{17}{11}}}$

Step 13.4

Combine the numerators over the common denominator.

$\frac{432 \cdot 5^{\frac{5}{11}}}{121} + \frac{- 750 - 800 \cdot 5^{\frac{11}{11}}}{121 \cdot 5^{\frac{17}{11}}}$

Step 13.5

Simplify the numerator.

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Step 13.5.1

Divide $11$ by $11$ .

$\frac{432 \cdot 5^{\frac{5}{11}}}{121} + \frac{- 750 - 800 \cdot 5^{1}}{121 \cdot 5^{\frac{17}{11}}}$

Step 13.5.2

Raise $5$ to the power of $1$ .

$\frac{432 \cdot 5^{\frac{5}{11}}}{121} + \frac{- 750 - 800 \cdot 5}{121 \cdot 5^{\frac{17}{11}}}$

Step 13.5.3

Multiply $- 800$ by $5$ .

$\frac{432 \cdot 5^{\frac{5}{11}}}{121} + \frac{- 750 - 4000}{121 \cdot 5^{\frac{17}{11}}}$

Step 13.5.4

Subtract $4000$ from $- 750$ .

$\frac{432 \cdot 5^{\frac{5}{11}}}{121} + \frac{- 4750}{121 \cdot 5^{\frac{17}{11}}}$

Step 13.6

Move the negative in front of the fraction.

$\frac{432 \cdot 5^{\frac{5}{11}}}{121} - \frac{4750}{121 \cdot 5^{\frac{17}{11}}}$

Step 14

$x = 5$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = 5$ is a local minimum

Step 15

Find the y-value when $x = 5$ .

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Step 15.1

Replace the variable $x$ with $5$ in the expression.

$f (5) = {(5)}^{\frac{5}{11}} {((5) - 5)}^{2}$

Step 15.2

Simplify the result.

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Step 15.2.1

Subtract $5$ from $5$ .

$f (5) = 5^{\frac{5}{11}} \cdot 0^{2}$

Step 15.2.2

Raising $0$ to any positive power yields $0$ .

$f (5) = 5^{\frac{5}{11}} \cdot 0$

Step 15.2.3

Multiply $5^{\frac{5}{11}}$ by $0$ .

$f (5) = 0$

Step 15.2.4

The final answer is $0$ .

$y = 0$

Step 16

Evaluate the second derivative at $x = 0$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{432 {(0)}^{\frac{5}{11}}}{121} - \frac{750}{121 {(0)}^{\frac{17}{11}}} - \frac{800}{121 {(0)}^{\frac{6}{11}}}$

Step 17

Evaluate the second derivative.

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Step 17.1

Simplify the expression.

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Step 17.1.1

Rewrite $0$ as $0^{11}$ .

$\frac{432 {(0)}^{\frac{5}{11}}}{121} - \frac{750}{121 {(0^{11})}^{\frac{17}{11}}} - \frac{800}{121 {(0)}^{\frac{6}{11}}}$

Step 17.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{432 {(0)}^{\frac{5}{11}}}{121} - \frac{750}{121 \cdot 0^{11 (\frac{17}{11})}} - \frac{800}{121 {(0)}^{\frac{6}{11}}}$

Step 17.2

Cancel the common factor of $11$ .

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Step 17.2.1

Cancel the common factor.

$\frac{432 {(0)}^{\frac{5}{11}}}{121} - \frac{750}{121 \cdot 0^{11 (\frac{17}{11})}} - \frac{800}{121 {(0)}^{\frac{6}{11}}}$

Step 17.2.2

Rewrite the expression.

$\frac{432 {(0)}^{\frac{5}{11}}}{121} - \frac{750}{121 \cdot 0^{17}} - \frac{800}{121 {(0)}^{\frac{6}{11}}}$

Step 17.3

Simplify the expression.

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Step 17.3.1

Raising $0$ to any positive power yields $0$ .

$\frac{432 {(0)}^{\frac{5}{11}}}{121} - \frac{750}{121 \cdot 0} - \frac{800}{121 {(0)}^{\frac{6}{11}}}$

Step 17.3.2

Multiply $121$ by $0$ .

$\frac{432 {(0)}^{\frac{5}{11}}}{121} - \frac{750}{0} - \frac{800}{121 {(0)}^{\frac{6}{11}}}$

Step 17.3.3

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{432 {(0)}^{\frac{5}{11}}}{121} - \frac{750}{0} - \frac{800}{121 {(0)}^{\frac{6}{11}}}$

Step 17.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

Step 18

Since there is at least one point with $0$ or undefined second derivative, apply the first derivative test.

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Step 18.1

Split $(- \infty, \infty)$ into separate intervals around the $x$ values that make the first derivative $0$ or undefined.

$(- \infty, 0) \cup (0, \frac{25}{27}) \cup (\frac{25}{27}, 5) \cup (5, \infty)$

Step 18.2

Substitute any number, such as $- 2$ , from the interval $(- \infty, 0)$ in the first derivative $\frac{27 x^{\frac{16}{11}}}{11} + \frac{125}{11 x^{\frac{6}{11}}} - \frac{160 x^{\frac{5}{11}}}{11}$ to check if the result is negative or positive.

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Step 18.2.1

Replace the variable $x$ with $- 2$ in the expression.

$f' (- 2) = \frac{27 {(- 2)}^{\frac{16}{11}}}{11} + \frac{125}{11 {(- 2)}^{\frac{6}{11}}} - \frac{160 {(- 2)}^{\frac{5}{11}}}{11}$

Step 18.2.2

Simplify the result.

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Step 18.2.2.1

Combine the numerators over the common denominator.

$f' (- 2) = \frac{27 {(- 2)}^{\frac{16}{11}} - 160 {(- 2)}^{\frac{5}{11}}}{11} + \frac{125}{11 {(- 2)}^{\frac{6}{11}}}$

Step 18.2.2.2

The final answer is $\frac{27 {(- 2)}^{\frac{16}{11}} - 160 {(- 2)}^{\frac{5}{11}}}{11} + \frac{125}{11 {(- 2)}^{\frac{6}{11}}}$ .

$\frac{27 {(- 2)}^{\frac{16}{11}} - 160 {(- 2)}^{\frac{5}{11}}}{11} + \frac{125}{11 {(- 2)}^{\frac{6}{11}}}$

Step 18.3

Substitute any number, such as $0.46$ , from the interval $(0, \frac{25}{27})$ in the first derivative $\frac{27 x^{\frac{16}{11}}}{11} + \frac{125}{11 x^{\frac{6}{11}}} - \frac{160 x^{\frac{5}{11}}}{11}$ to check if the result is negative or positive.

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Step 18.3.1

Replace the variable $x$ with $0.46$ in the expression.

$f' (0.46) = \frac{27 {(0.46)}^{\frac{16}{11}}}{11} + \frac{125}{11 {(0.46)}^{\frac{6}{11}}} - \frac{160 {(0.46)}^{\frac{5}{11}}}{11}$

Step 18.3.2

Simplify the result.

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Step 18.3.2.1

Combine the numerators over the common denominator.

$f' (0.46) = \frac{27 {(0.46)}^{\frac{16}{11}} - 160 {(0.46)}^{\frac{5}{11}}}{11} + \frac{125}{11 {(0.46)}^{\frac{6}{11}}}$

Step 18.3.2.2

Simplify each term.

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Step 18.3.2.2.1

Raise $0.46$ to the power of $\frac{16}{11}$ .

$f' (0.46) = \frac{27 \cdot 0.32319597 - 160 {(0.46)}^{\frac{5}{11}}}{11} + \frac{125}{11 {(0.46)}^{\frac{6}{11}}}$

Step 18.3.2.2.2

Multiply $27$ by $0.32319597$ .

$f' (0.46) = \frac{8.72629119 - 160 {(0.46)}^{\frac{5}{11}}}{11} + \frac{125}{11 {(0.46)}^{\frac{6}{11}}}$

Step 18.3.2.2.3

Raise $0.46$ to the power of $\frac{5}{11}$ .

$f' (0.46) = \frac{8.72629119 - 160 \cdot 0.70259993}{11} + \frac{125}{11 {(0.46)}^{\frac{6}{11}}}$

Step 18.3.2.2.4

Multiply $- 160$ by $0.70259993$ .

$f' (0.46) = \frac{8.72629119 - 112.4159896}{11} + \frac{125}{11 {(0.46)}^{\frac{6}{11}}}$

Step 18.3.2.3

Subtract $112.4159896$ from $8.72629119$ .

$f' (0.46) = \frac{- 103.68969841}{11} + \frac{125}{11 {(0.46)}^{\frac{6}{11}}}$

Step 18.3.2.4

Simplify each term.

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Step 18.3.2.4.1

Divide $- 103.68969841$ by $11$ .

$f' (0.46) = - 9.42633621 + \frac{125}{11 {(0.46)}^{\frac{6}{11}}}$

Step 18.3.2.4.2

Raise $0.46$ to the power of $\frac{6}{11}$ .

$f' (0.46) = - 9.42633621 + \frac{125}{11 \cdot 0.65471113}$

Step 18.3.2.4.3

Multiply $11$ by $0.65471113$ .

$f' (0.46) = - 9.42633621 + \frac{125}{7.20182247}$

Step 18.3.2.4.4

Divide $125$ by $7.20182247$ .

$f' (0.46) = - 9.42633621 + 17.35671776$

Step 18.3.2.5

Add $- 9.42633621$ and $17.35671776$ .

$f' (0.46) = 7.93038154$

Step 18.3.2.6

The final answer is $7.93038154$ .

$7.93038154$

Step 18.4

Substitute any number, such as $3$ , from the interval $(\frac{25}{27}, 5)$ in the first derivative $\frac{27 x^{\frac{16}{11}}}{11} + \frac{125}{11 x^{\frac{6}{11}}} - \frac{160 x^{\frac{5}{11}}}{11}$ to check if the result is negative or positive.

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Step 18.4.1

Replace the variable $x$ with $3$ in the expression.

$f' (3) = \frac{27 {(3)}^{\frac{16}{11}}}{11} + \frac{125}{11 {(3)}^{\frac{6}{11}}} - \frac{160 {(3)}^{\frac{5}{11}}}{11}$

Step 18.4.2

Simplify the result.

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Step 18.4.2.1

Simplify the numerator.

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Step 18.4.2.1.1

Rewrite $27$ as $3^{3}$ .

$f' (3) = \frac{3^{3} \cdot 3^{\frac{16}{11}}}{11} + \frac{125}{11 {(3)}^{\frac{6}{11}}} - \frac{160 {(3)}^{\frac{5}{11}}}{11}$

Step 18.4.2.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f' (3) = \frac{3^{3 + \frac{16}{11}}}{11} + \frac{125}{11 {(3)}^{\frac{6}{11}}} - \frac{160 {(3)}^{\frac{5}{11}}}{11}$

Step 18.4.2.1.3

To write $3$ as a fraction with a common denominator, multiply by $\frac{11}{11}$ .

$f' (3) = \frac{3^{3 \cdot \frac{11}{11} + \frac{16}{11}}}{11} + \frac{125}{11 {(3)}^{\frac{6}{11}}} - \frac{160 {(3)}^{\frac{5}{11}}}{11}$

Step 18.4.2.1.4

Combine $3$ and $\frac{11}{11}$ .

$f' (3) = \frac{3^{\frac{3 \cdot 11}{11} + \frac{16}{11}}}{11} + \frac{125}{11 {(3)}^{\frac{6}{11}}} - \frac{160 {(3)}^{\frac{5}{11}}}{11}$

Step 18.4.2.1.5

Combine the numerators over the common denominator.

$f' (3) = \frac{3^{\frac{3 \cdot 11 + 16}{11}}}{11} + \frac{125}{11 {(3)}^{\frac{6}{11}}} - \frac{160 {(3)}^{\frac{5}{11}}}{11}$

Step 18.4.2.1.6

Simplify the numerator.

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Step 18.4.2.1.6.1

Multiply $3$ by $11$ .

$f' (3) = \frac{3^{\frac{33 + 16}{11}}}{11} + \frac{125}{11 {(3)}^{\frac{6}{11}}} - \frac{160 {(3)}^{\frac{5}{11}}}{11}$

Step 18.4.2.1.6.2

Add $33$ and $16$ .

$f' (3) = \frac{3^{\frac{49}{11}}}{11} + \frac{125}{11 {(3)}^{\frac{6}{11}}} - \frac{160 {(3)}^{\frac{5}{11}}}{11}$

$f' (3) = \frac{3^{\frac{49}{11}}}{11} + \frac{125}{11 \cdot 3^{\frac{6}{11}}} - \frac{160 \cdot 3^{\frac{5}{11}}}{11}$

Step 18.4.2.2

Combine the numerators over the common denominator.

$f' (3) = \frac{3^{\frac{49}{11}} - 160 \cdot 3^{\frac{5}{11}}}{11} + \frac{125}{11 \cdot 3^{\frac{6}{11}}}$

Step 18.4.2.3

The final answer is $\frac{3^{\frac{49}{11}} - 160 \cdot 3^{\frac{5}{11}}}{11} + \frac{125}{11 \cdot 3^{\frac{6}{11}}}$ .

$\frac{3^{\frac{49}{11}} - 160 \cdot 3^{\frac{5}{11}}}{11} + \frac{125}{11 \cdot 3^{\frac{6}{11}}}$

Step 18.5

Substitute any number, such as $8$ , from the interval $(5, \infty)$ in the first derivative $\frac{27 x^{\frac{16}{11}}}{11} + \frac{125}{11 x^{\frac{6}{11}}} - \frac{160 x^{\frac{5}{11}}}{11}$ to check if the result is negative or positive.

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Step 18.5.1

Replace the variable $x$ with $8$ in the expression.

$f' (8) = \frac{27 {(8)}^{\frac{16}{11}}}{11} + \frac{125}{11 {(8)}^{\frac{6}{11}}} - \frac{160 {(8)}^{\frac{5}{11}}}{11}$

Step 18.5.2

Simplify the result.

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Step 18.5.2.1

Remove parentheses.

$f' (8) = \frac{27 \cdot 8^{\frac{16}{11}}}{11} + \frac{125}{11 \cdot 8^{\frac{6}{11}}} - \frac{160 \cdot 8^{\frac{5}{11}}}{11}$

Step 18.5.2.2

Combine the numerators over the common denominator.

$f' (8) = \frac{27 \cdot 8^{\frac{16}{11}} - 160 \cdot 8^{\frac{5}{11}}}{11} + \frac{125}{11 \cdot 8^{\frac{6}{11}}}$

Step 18.5.2.3

The final answer is $\frac{27 \cdot 8^{\frac{16}{11}} - 160 \cdot 8^{\frac{5}{11}}}{11} + \frac{125}{11 \cdot 8^{\frac{6}{11}}}$ .

$\frac{27 \cdot 8^{\frac{16}{11}} - 160 \cdot 8^{\frac{5}{11}}}{11} + \frac{125}{11 \cdot 8^{\frac{6}{11}}}$

Step 18.6

Since the first derivative did not change signs around $x = 0$ , this is not a local maximum or minimum.

Not a local maximum or minimum

Step 18.7

Since the first derivative changed signs from positive to negative around $x = \frac{25}{27}$ , then $x = \frac{25}{27}$ is a local maximum.

$x = \frac{25}{27}$ is a local maximum

Step 18.8

Since the first derivative changed signs from negative to positive around $x = 5$ , then $x = 5$ is a local minimum.

$x = 5$ is a local minimum

Step 18.9

These are the local extrema for $f (x) = x^{\frac{5}{11}} {(x - 5)}^{2}$ .

$x = \frac{25}{27}$ is a local maximum

$x = 5$ is a local minimum

$x = \frac{25}{27}$ is a local maximum

$x = 5$ is a local minimum

Step 19