Calculus Examples

$f (x) = x^{2} - 32 \sqrt{x} - 1$

Step 1

Find the first derivative of the function.

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Step 1.1

Differentiate.

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Step 1.1.1

By the Sum Rule, the derivative of $x^{2} - 32 \sqrt{x} - 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 32 \sqrt{x}] + \frac{d}{d x} [- 1]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 32 \sqrt{x}] + \frac{d}{d x} [- 1]$

Step 1.1.2

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

$2 x + \frac{d}{d x} [- 32 \sqrt{x}] + \frac{d}{d x} [- 1]$

Step 1.2

Evaluate $\frac{d}{d x} [- 32 \sqrt{x}]$ .

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Step 1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$2 x + \frac{d}{d x} [- 32 x^{\frac{1}{2}}] + \frac{d}{d x} [- 1]$

Step 1.2.2

Since $- 32$ is constant with respect to $x$ , the derivative of $- 32 x^{\frac{1}{2}}$ with respect to $x$ is $- 32 \frac{d}{d x} [x^{\frac{1}{2}}]$ .

$2 x - 32 \frac{d}{d x} [x^{\frac{1}{2}}] + \frac{d}{d x} [- 1]$

Step 1.2.3

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

$2 x - 32 (\frac{1}{2} x^{\frac{1}{2} - 1}) + \frac{d}{d x} [- 1]$

Step 1.2.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$2 x - 32 (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}}) + \frac{d}{d x} [- 1]$

Step 1.2.5

Combine $- 1$ and $\frac{2}{2}$ .

$2 x - 32 (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}) + \frac{d}{d x} [- 1]$

Step 1.2.6

Combine the numerators over the common denominator.

$2 x - 32 (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}}) + \frac{d}{d x} [- 1]$

Step 1.2.7

Simplify the numerator.

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Step 1.2.7.1

Multiply $- 1$ by $2$ .

$2 x - 32 (\frac{1}{2} x^{\frac{1 - 2}{2}}) + \frac{d}{d x} [- 1]$

Step 1.2.7.2

Subtract $2$ from $1$ .

$2 x - 32 (\frac{1}{2} x^{\frac{- 1}{2}}) + \frac{d}{d x} [- 1]$

Step 1.2.8

Move the negative in front of the fraction.

$2 x - 32 (\frac{1}{2} x^{- \frac{1}{2}}) + \frac{d}{d x} [- 1]$

Step 1.2.9

Combine $\frac{1}{2}$ and $x^{- \frac{1}{2}}$ .

$2 x - 32 \frac{x^{- \frac{1}{2}}}{2} + \frac{d}{d x} [- 1]$

Step 1.2.10

Combine $- 32$ and $\frac{x^{- \frac{1}{2}}}{2}$ .

$2 x + \frac{- 32 x^{- \frac{1}{2}}}{2} + \frac{d}{d x} [- 1]$

Step 1.2.11

Move $x^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$2 x + \frac{- 32}{2 x^{\frac{1}{2}}} + \frac{d}{d x} [- 1]$

Step 1.2.12

Factor $2$ out of $- 32$ .

$2 x + \frac{2 \cdot - 16}{2 x^{\frac{1}{2}}} + \frac{d}{d x} [- 1]$

Step 1.2.13

Cancel the common factors.

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Step 1.2.13.1

Factor $2$ out of $2 x^{\frac{1}{2}}$ .

$2 x + \frac{2 \cdot - 16}{2 (x^{\frac{1}{2}})} + \frac{d}{d x} [- 1]$

Step 1.2.13.2

Cancel the common factor.

$2 x + \frac{2 \cdot - 16}{2 x^{\frac{1}{2}}} + \frac{d}{d x} [- 1]$

Step 1.2.13.3

Rewrite the expression.

$2 x + \frac{- 16}{x^{\frac{1}{2}}} + \frac{d}{d x} [- 1]$

Step 1.2.14

Move the negative in front of the fraction.

$2 x - \frac{16}{x^{\frac{1}{2}}} + \frac{d}{d x} [- 1]$

Step 1.3

Differentiate using the Constant Rule.

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Step 1.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$2 x - \frac{16}{x^{\frac{1}{2}}} + 0$

Step 1.3.2

Add $2 x - \frac{16}{x^{\frac{1}{2}}}$ and $0$ .

$2 x - \frac{16}{x^{\frac{1}{2}}}$

Step 2

Find the second derivative of the function.

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Step 2.1

By the Sum Rule, the derivative of $2 x - \frac{16}{x^{\frac{1}{2}}}$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [- \frac{16}{x^{\frac{1}{2}}}]$ .

$f'' (x) = \frac{d}{d x} (2 x) + \frac{d}{d x} (- \frac{16}{x^{\frac{1}{2}}})$

Step 2.2

Evaluate $\frac{d}{d x} [2 x]$ .

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Step 2.2.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$f'' (x) = 2 \frac{d}{d x} (x) + \frac{d}{d x} (- \frac{16}{x^{\frac{1}{2}}})$

Step 2.2.2

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

$f'' (x) = 2 \cdot 1 + \frac{d}{d x} (- \frac{16}{x^{\frac{1}{2}}})$

Step 2.2.3

Multiply $2$ by $1$ .

$f'' (x) = 2 + \frac{d}{d x} (- \frac{16}{x^{\frac{1}{2}}})$

Step 2.3

Evaluate $\frac{d}{d x} [- \frac{16}{x^{\frac{1}{2}}}]$ .

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Step 2.3.1

Since $- 16$ is constant with respect to $x$ , the derivative of $- \frac{16}{x^{\frac{1}{2}}}$ with respect to $x$ is $- 16 \frac{d}{d x} [\frac{1}{x^{\frac{1}{2}}}]$ .

$f'' (x) = 2 - 16 \frac{d}{d x} \frac{1}{x^{\frac{1}{2}}}$

Step 2.3.2

Rewrite $\frac{1}{x^{\frac{1}{2}}}$ as ${(x^{\frac{1}{2}})}^{- 1}$ .

$f'' (x) = 2 - 16 \frac{d}{d x} {(x^{\frac{1}{2}})}^{- 1}$

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 $g (x) = x^{\frac{1}{2}}$ .

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Step 2.3.3.1

To apply the Chain Rule, set $u$ as $x^{\frac{1}{2}}$ .

$f'' (x) = 2 - 16 (\frac{d}{d u} (u^{- 1}) \frac{d}{d x} (x^{\frac{1}{2}}))$

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 $n = - 1$ .

$f'' (x) = 2 - 16 (- u^{- 2} \frac{d}{d x} x^{\frac{1}{2}})$

Step 2.3.3.3

Replace all occurrences of $u$ with $x^{\frac{1}{2}}$ .

$f'' (x) = 2 - 16 (- {(x^{\frac{1}{2}})}^{- 2} \frac{d}{d x} x^{\frac{1}{2}})$

Step 2.3.4

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

$f'' (x) = 2 - 16 (- {(x^{\frac{1}{2}})}^{- 2} (\frac{1}{2} x^{\frac{1}{2} - 1}))$

Step 2.3.5

Multiply the exponents in ${(x^{\frac{1}{2}})}^{- 2}$ .

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Step 2.3.5.1

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

$f'' (x) = 2 - 16 (- x^{\frac{1}{2} \cdot - 2} (\frac{1}{2} x^{\frac{1}{2} - 1}))$

Step 2.3.5.2

Cancel the common factor of $2$ .

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Step 2.3.5.2.1

Factor $2$ out of $- 2$ .

$f'' (x) = 2 - 16 (- x^{\frac{1}{2} \cdot (2 (- 1))} (\frac{1}{2} x^{\frac{1}{2} - 1}))$

Step 2.3.5.2.2

Cancel the common factor.

$f'' (x) = 2 - 16 (- x^{\frac{1}{2} \cdot (2 \cdot - 1)} (\frac{1}{2} x^{\frac{1}{2} - 1}))$

Step 2.3.5.2.3

Rewrite the expression.

$f'' (x) = 2 - 16 (- x^{- 1} (\frac{1}{2} x^{\frac{1}{2} - 1}))$

Step 2.3.6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f'' (x) = 2 - 16 (- x^{- 1} (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}}))$

Step 2.3.7

Combine $- 1$ and $\frac{2}{2}$ .

$f'' (x) = 2 - 16 (- x^{- 1} (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}))$

Step 2.3.8

Combine the numerators over the common denominator.

$f'' (x) = 2 - 16 (- x^{- 1} (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}}))$

Step 2.3.9

Simplify the numerator.

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Step 2.3.9.1

Multiply $- 1$ by $2$ .

$f'' (x) = 2 - 16 (- x^{- 1} (\frac{1}{2} x^{\frac{1 - 2}{2}}))$

Step 2.3.9.2

Subtract $2$ from $1$ .

$f'' (x) = 2 - 16 (- x^{- 1} (\frac{1}{2} x^{\frac{- 1}{2}}))$

Step 2.3.10

Move the negative in front of the fraction.

$f'' (x) = 2 - 16 (- x^{- 1} (\frac{1}{2} x^{- \frac{1}{2}}))$

Step 2.3.11

Combine $\frac{1}{2}$ and $x^{- \frac{1}{2}}$ .

$f'' (x) = 2 - 16 (- x^{- 1} \frac{x^{- \frac{1}{2}}}{2})$

Step 2.3.12

Combine $\frac{x^{- \frac{1}{2}}}{2}$ and $x^{- 1}$ .

$f'' (x) = 2 - 16 (- \frac{x^{- \frac{1}{2}} x^{- 1}}{2})$

Step 2.3.13

Multiply $x^{- \frac{1}{2}}$ by $x^{- 1}$ by adding the exponents.

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Step 2.3.13.1

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

$f'' (x) = 2 - 16 (- \frac{x^{- \frac{1}{2} - 1}}{2})$

Step 2.3.13.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f'' (x) = 2 - 16 (- \frac{x^{- \frac{1}{2} - 1 \cdot \frac{2}{2}}}{2})$

Step 2.3.13.3

Combine $- 1$ and $\frac{2}{2}$ .

$f'' (x) = 2 - 16 (- \frac{x^{- \frac{1}{2} + \frac{- 1 \cdot 2}{2}}}{2})$

Step 2.3.13.4

Combine the numerators over the common denominator.

$f'' (x) = 2 - 16 (- \frac{x^{\frac{- 1 - 1 \cdot 2}{2}}}{2})$

Step 2.3.13.5

Simplify the numerator.

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Step 2.3.13.5.1

Multiply $- 1$ by $2$ .

$f'' (x) = 2 - 16 (- \frac{x^{\frac{- 1 - 2}{2}}}{2})$

Step 2.3.13.5.2

Subtract $2$ from $- 1$ .

$f'' (x) = 2 - 16 (- \frac{x^{\frac{- 3}{2}}}{2})$

Step 2.3.13.6

Move the negative in front of the fraction.

$f'' (x) = 2 - 16 (- \frac{x^{- \frac{3}{2}}}{2})$

Step 2.3.14

Move $x^{- \frac{3}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (x) = 2 - 16 (- \frac{1}{2 x^{\frac{3}{2}}})$

Step 2.3.15

Multiply $- 1$ by $- 16$ .

$f'' (x) = 2 + 16 (\frac{1}{2 x^{\frac{3}{2}}})$

Step 2.3.16

Combine $16$ and $\frac{1}{2 x^{\frac{3}{2}}}$ .

$f'' (x) = 2 + \frac{16}{2 x^{\frac{3}{2}}}$

Step 2.3.17

Factor $2$ out of $16$ .

$f'' (x) = 2 + \frac{2 \cdot 8}{2 x^{\frac{3}{2}}}$

Step 2.3.18

Cancel the common factors.

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Step 2.3.18.1

Factor $2$ out of $2 x^{\frac{3}{2}}$ .

$f'' (x) = 2 + \frac{2 \cdot 8}{2 (x^{\frac{3}{2}})}$

Step 2.3.18.2

Cancel the common factor.

$f'' (x) = 2 + \frac{2 \cdot 8}{2 x^{\frac{3}{2}}}$

Step 2.3.18.3

Rewrite the expression.

$f'' (x) = 2 + \frac{8}{x^{\frac{3}{2}}}$

Step 3

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

$2 x - \frac{16}{x^{\frac{1}{2}}} = 0$

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

Differentiate.

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Step 4.1.1.1

By the Sum Rule, the derivative of $x^{2} - 32 \sqrt{x} - 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 32 \sqrt{x}] + \frac{d}{d x} [- 1]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 32 \sqrt{x}] + \frac{d}{d x} [- 1]$

Step 4.1.1.2

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

$2 x + \frac{d}{d x} [- 32 \sqrt{x}] + \frac{d}{d x} [- 1]$

Step 4.1.2

Evaluate $\frac{d}{d x} [- 32 \sqrt{x}]$ .

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Step 4.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$2 x + \frac{d}{d x} [- 32 x^{\frac{1}{2}}] + \frac{d}{d x} [- 1]$

Step 4.1.2.2

Since $- 32$ is constant with respect to $x$ , the derivative of $- 32 x^{\frac{1}{2}}$ with respect to $x$ is $- 32 \frac{d}{d x} [x^{\frac{1}{2}}]$ .

$2 x - 32 \frac{d}{d x} [x^{\frac{1}{2}}] + \frac{d}{d x} [- 1]$

Step 4.1.2.3

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

$2 x - 32 (\frac{1}{2} x^{\frac{1}{2} - 1}) + \frac{d}{d x} [- 1]$

Step 4.1.2.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$2 x - 32 (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}}) + \frac{d}{d x} [- 1]$

Step 4.1.2.5

Combine $- 1$ and $\frac{2}{2}$ .

$2 x - 32 (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}) + \frac{d}{d x} [- 1]$

Step 4.1.2.6

Combine the numerators over the common denominator.

$2 x - 32 (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}}) + \frac{d}{d x} [- 1]$

Step 4.1.2.7

Simplify the numerator.

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Step 4.1.2.7.1

Multiply $- 1$ by $2$ .

$2 x - 32 (\frac{1}{2} x^{\frac{1 - 2}{2}}) + \frac{d}{d x} [- 1]$

Step 4.1.2.7.2

Subtract $2$ from $1$ .

$2 x - 32 (\frac{1}{2} x^{\frac{- 1}{2}}) + \frac{d}{d x} [- 1]$

Step 4.1.2.8

Move the negative in front of the fraction.

$2 x - 32 (\frac{1}{2} x^{- \frac{1}{2}}) + \frac{d}{d x} [- 1]$

Step 4.1.2.9

Combine $\frac{1}{2}$ and $x^{- \frac{1}{2}}$ .

$2 x - 32 \frac{x^{- \frac{1}{2}}}{2} + \frac{d}{d x} [- 1]$

Step 4.1.2.10

Combine $- 32$ and $\frac{x^{- \frac{1}{2}}}{2}$ .

$2 x + \frac{- 32 x^{- \frac{1}{2}}}{2} + \frac{d}{d x} [- 1]$

Step 4.1.2.11

Move $x^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$2 x + \frac{- 32}{2 x^{\frac{1}{2}}} + \frac{d}{d x} [- 1]$

Step 4.1.2.12

Factor $2$ out of $- 32$ .

$2 x + \frac{2 \cdot - 16}{2 x^{\frac{1}{2}}} + \frac{d}{d x} [- 1]$

Step 4.1.2.13

Cancel the common factors.

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Step 4.1.2.13.1

Factor $2$ out of $2 x^{\frac{1}{2}}$ .

$2 x + \frac{2 \cdot - 16}{2 (x^{\frac{1}{2}})} + \frac{d}{d x} [- 1]$

Step 4.1.2.13.2

Cancel the common factor.

$2 x + \frac{2 \cdot - 16}{2 x^{\frac{1}{2}}} + \frac{d}{d x} [- 1]$

Step 4.1.2.13.3

Rewrite the expression.

$2 x + \frac{- 16}{x^{\frac{1}{2}}} + \frac{d}{d x} [- 1]$

Step 4.1.2.14

Move the negative in front of the fraction.

$2 x - \frac{16}{x^{\frac{1}{2}}} + \frac{d}{d x} [- 1]$

Step 4.1.3

Differentiate using the Constant Rule.

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Step 4.1.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$2 x - \frac{16}{x^{\frac{1}{2}}} + 0$

Step 4.1.3.2

Add $2 x - \frac{16}{x^{\frac{1}{2}}}$ and $0$ .

$f' (x) = 2 x - \frac{16}{x^{\frac{1}{2}}}$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $2 x - \frac{16}{x^{\frac{1}{2}}}$ .

$2 x - \frac{16}{x^{\frac{1}{2}}}$

Step 5

Set the first derivative equal to $0$ then solve the equation $2 x - \frac{16}{x^{\frac{1}{2}}} = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$2 x - \frac{16}{x^{\frac{1}{2}}} = 0$

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.

$1, x^{\frac{1}{2}}, 1$

Step 5.2.2

The LCM of one and any expression is the expression.

$x^{\frac{1}{2}}$

Step 5.3

Multiply each term in $2 x - \frac{16}{x^{\frac{1}{2}}} = 0$ by $x^{\frac{1}{2}}$ to eliminate the fractions.

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Step 5.3.1

Multiply each term in $2 x - \frac{16}{x^{\frac{1}{2}}} = 0$ by $x^{\frac{1}{2}}$ .

$2 x \cdot x^{\frac{1}{2}} - \frac{16}{x^{\frac{1}{2}}} x^{\frac{1}{2}} = 0 x^{\frac{1}{2}}$

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

Multiply $x$ by $x^{\frac{1}{2}}$ by adding the exponents.

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Step 5.3.2.1.1.1

Move $x^{\frac{1}{2}}$ .

$2 (x^{\frac{1}{2}} x) - \frac{16}{x^{\frac{1}{2}}} x^{\frac{1}{2}} = 0 x^{\frac{1}{2}}$

Step 5.3.2.1.1.2

Multiply $x^{\frac{1}{2}}$ by $x$ .

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Step 5.3.2.1.1.2.1

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

$2 (x^{\frac{1}{2}} x^{1}) - \frac{16}{x^{\frac{1}{2}}} x^{\frac{1}{2}} = 0 x^{\frac{1}{2}}$

Step 5.3.2.1.1.2.2

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

$2 x^{\frac{1}{2} + 1} - \frac{16}{x^{\frac{1}{2}}} x^{\frac{1}{2}} = 0 x^{\frac{1}{2}}$

Step 5.3.2.1.1.3

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

$2 x^{\frac{1}{2} + \frac{2}{2}} - \frac{16}{x^{\frac{1}{2}}} x^{\frac{1}{2}} = 0 x^{\frac{1}{2}}$

Step 5.3.2.1.1.4

Combine the numerators over the common denominator.

$2 x^{\frac{1 + 2}{2}} - \frac{16}{x^{\frac{1}{2}}} x^{\frac{1}{2}} = 0 x^{\frac{1}{2}}$

Step 5.3.2.1.1.5

Add $1$ and $2$ .

$2 x^{\frac{3}{2}} - \frac{16}{x^{\frac{1}{2}}} x^{\frac{1}{2}} = 0 x^{\frac{1}{2}}$

Step 5.3.2.1.2

Cancel the common factor of $x^{\frac{1}{2}}$ .

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Step 5.3.2.1.2.1

Move the leading negative in $- \frac{16}{x^{\frac{1}{2}}}$ into the numerator.

$2 x^{\frac{3}{2}} + \frac{- 16}{x^{\frac{1}{2}}} x^{\frac{1}{2}} = 0 x^{\frac{1}{2}}$

Step 5.3.2.1.2.2

Cancel the common factor.

$2 x^{\frac{3}{2}} + \frac{- 16}{x^{\frac{1}{2}}} x^{\frac{1}{2}} = 0 x^{\frac{1}{2}}$

Step 5.3.2.1.2.3

Rewrite the expression.

$2 x^{\frac{3}{2}} - 16 = 0 x^{\frac{1}{2}}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Multiply $0$ by $x^{\frac{1}{2}}$ .

$2 x^{\frac{3}{2}} - 16 = 0$

Step 5.4

Solve the equation.

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Step 5.4.1

Add $16$ to both sides of the equation.

$2 x^{\frac{3}{2}} = 16$

Step 5.4.2

Raise each side of the equation to the power of $\frac{2}{3}$ to eliminate the fractional exponent on the left side.

${(2 x^{\frac{3}{2}})}^{\frac{2}{3}} = 16^{\frac{2}{3}}$

Step 5.4.3

Simplify the left side.

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Step 5.4.3.1

Simplify ${(2 x^{\frac{3}{2}})}^{\frac{2}{3}}$ .

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Step 5.4.3.1.1

Apply the product rule to $2 x^{\frac{3}{2}}$ .

$2^{\frac{2}{3}} {(x^{\frac{3}{2}})}^{\frac{2}{3}} = 16^{\frac{2}{3}}$

Step 5.4.3.1.2

Multiply the exponents in ${(x^{\frac{3}{2}})}^{\frac{2}{3}}$ .

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Step 5.4.3.1.2.1

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

$2^{\frac{2}{3}} x^{\frac{3}{2} \cdot \frac{2}{3}} = 16^{\frac{2}{3}}$

Step 5.4.3.1.2.2

Cancel the common factor of $3$ .

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Step 5.4.3.1.2.2.1

Cancel the common factor.

$2^{\frac{2}{3}} x^{\frac{3}{2} \cdot \frac{2}{3}} = 16^{\frac{2}{3}}$

Step 5.4.3.1.2.2.2

Rewrite the expression.

$2^{\frac{2}{3}} x^{\frac{1}{2} \cdot 2} = 16^{\frac{2}{3}}$

Step 5.4.3.1.2.3

Cancel the common factor of $2$ .

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Step 5.4.3.1.2.3.1

Cancel the common factor.

$2^{\frac{2}{3}} x^{\frac{1}{2} \cdot 2} = 16^{\frac{2}{3}}$

Step 5.4.3.1.2.3.2

Rewrite the expression.

$2^{\frac{2}{3}} x^{1} = 16^{\frac{2}{3}}$

Step 5.4.3.1.3

Simplify.

$2^{\frac{2}{3}} x = 16^{\frac{2}{3}}$

Step 5.4.3.1.4

Reorder factors in $2^{\frac{2}{3}} x$ .

$x \cdot 2^{\frac{2}{3}} = 16^{\frac{2}{3}}$

Step 5.4.4

Divide each term in $x \cdot 2^{\frac{2}{3}} = 16^{\frac{2}{3}}$ by $2^{\frac{2}{3}}$ and simplify.

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Step 5.4.4.1

Divide each term in $x \cdot 2^{\frac{2}{3}} = 16^{\frac{2}{3}}$ by $2^{\frac{2}{3}}$ .

$\frac{x \cdot 2^{\frac{2}{3}}}{2^{\frac{2}{3}}} = \frac{16^{\frac{2}{3}}}{2^{\frac{2}{3}}}$

Step 5.4.4.2

Simplify the left side.

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Step 5.4.4.2.1

Cancel the common factor.

$\frac{x \cdot 2^{\frac{2}{3}}}{2^{\frac{2}{3}}} = \frac{16^{\frac{2}{3}}}{2^{\frac{2}{3}}}$

Step 5.4.4.2.2

Divide $x$ by $1$ .

$x = \frac{16^{\frac{2}{3}}}{2^{\frac{2}{3}}}$

Step 5.4.4.3

Simplify the right side.

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Step 5.4.4.3.1

Use the power of quotient rule $\frac{a^{m}}{b^{m}} = {(\frac{a}{b})}^{m}$ .

$x = {(\frac{16}{2})}^{\frac{2}{3}}$

Step 5.4.4.3.2

Simplify the expression.

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Step 5.4.4.3.2.1

Divide $16$ by $2$ .

$x = 8^{\frac{2}{3}}$

Step 5.4.4.3.2.2

Rewrite $8$ as $2^{3}$ .

$x = {(2^{3})}^{\frac{2}{3}}$

Step 5.4.4.3.2.3

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

$x = 2^{3 (\frac{2}{3})}$

Step 5.4.4.3.3

Cancel the common factor of $3$ .

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Step 5.4.4.3.3.1

Cancel the common factor.

$x = 2^{3 (\frac{2}{3})}$

Step 5.4.4.3.3.2

Rewrite the expression.

$x = 2^{2}$

Step 5.4.4.3.4

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

$x = 4$

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.

$2 x - \frac{16}{\sqrt{x^{1}}}$

Step 6.1.2

Anything raised to $1$ is the base itself.

$2 x - \frac{16}{\sqrt{x}}$

Step 6.2

Set the denominator in $\frac{16}{\sqrt{x}}$ equal to $0$ to find where the expression is undefined.

$\sqrt{x} = 0$

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, square both sides of the equation.

${\sqrt{x}}^{2} = 0^{2}$

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 $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

${(x^{\frac{1}{2}})}^{2} = 0^{2}$

Step 6.3.2.2

Simplify the left side.

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Step 6.3.2.2.1

Simplify ${(x^{\frac{1}{2}})}^{2}$ .

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Step 6.3.2.2.1.1

Multiply the exponents in ${(x^{\frac{1}{2}})}^{2}$ .

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Step 6.3.2.2.1.1.1

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

$x^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 6.3.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 6.3.2.2.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 6.3.2.2.1.1.2.2

Rewrite the expression.

$x^{1} = 0^{2}$

Step 6.3.2.2.1.2

Simplify.

$x = 0^{2}$

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 $0$ .

$x = 0$

Step 6.4

Set the radicand in $\sqrt{x}$ less than $0$ to find where the expression is undefined.

$x < 0$

Step 6.5

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x \leq 0$

$(- \infty, 0]$

$x \leq 0$

$(- \infty, 0]$

Step 7

Critical points to evaluate.

$x = 4, 0$

Step 8

Evaluate the second derivative at $x = 4$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$2 + \frac{8}{{(4)}^{\frac{3}{2}}}$

Step 9

Evaluate the second derivative.

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Step 9.1

Simplify each term.

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Step 9.1.1

Simplify the denominator.

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Step 9.1.1.1

Rewrite $4$ as $2^{2}$ .

$2 + \frac{8}{{(2^{2})}^{\frac{3}{2}}}$

Step 9.1.1.2

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

$2 + \frac{8}{2^{2 (\frac{3}{2})}}$

Step 9.1.1.3

Cancel the common factor of $2$ .

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Step 9.1.1.3.1

Cancel the common factor.

$2 + \frac{8}{2^{2 (\frac{3}{2})}}$

Step 9.1.1.3.2

Rewrite the expression.

$2 + \frac{8}{2^{3}}$

Step 9.1.1.4

Raise $2$ to the power of $3$ .

$2 + \frac{8}{8}$

Step 9.1.2

Divide $8$ by $8$ .

$2 + 1$

Step 9.2

Add $2$ and $1$ .

$3$

Step 10

$x = 4$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = 4$ is a local minimum

Step 11

Find the y-value when $x = 4$ .

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Step 11.1

Replace the variable $x$ with $4$ in the expression.

$f (4) = {(4)}^{2} - 32 \sqrt{4} - 1$

Step 11.2

Simplify the result.

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Step 11.2.1

Simplify each term.

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Step 11.2.1.1

Raise $4$ to the power of $2$ .

$f (4) = 16 - 32 \sqrt{4} - 1$

Step 11.2.1.2

Rewrite $4$ as $2^{2}$ .

$f (4) = 16 - 32 \sqrt{2^{2}} - 1$

Step 11.2.1.3

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

$f (4) = 16 - 32 \cdot 2 - 1$

Step 11.2.1.4

Multiply $- 32$ by $2$ .

$f (4) = 16 - 64 - 1$

Step 11.2.2

Simplify by subtracting numbers.

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Step 11.2.2.1

Subtract $64$ from $16$ .

$f (4) = - 48 - 1$

Step 11.2.2.2

Subtract $1$ from $- 48$ .

$f (4) = - 49$

Step 11.2.3

The final answer is $- 49$ .

$y = - 49$

Step 12

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.

$2 + \frac{8}{{(0)}^{\frac{3}{2}}}$

Step 13

Evaluate the second derivative.

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Step 13.1

Simplify the expression.

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Step 13.1.1

Rewrite $0$ as $0^{2}$ .

$2 + \frac{8}{{(0^{2})}^{\frac{3}{2}}}$

Step 13.1.2

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

$2 + \frac{8}{0^{2 (\frac{3}{2})}}$

Step 13.2

Cancel the common factor of $2$ .

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Step 13.2.1

Cancel the common factor.

$2 + \frac{8}{0^{2 (\frac{3}{2})}}$

Step 13.2.2

Rewrite the expression.

$2 + \frac{8}{0^{3}}$

Step 13.3

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

$2 + \frac{8}{0}$

Step 13.4

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

Undefined

Step 14

Since the first derivative test failed, there are no local extrema.

No Local Extrema

Step 15