Calculus Examples

$f (x) = 9 x^{\frac{7}{5}} - 5 x^{2} + 10^{4}$

Step 1

Find the first derivative of the function.

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Step 1.1

Differentiate using the Sum Rule.

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Step 1.1.1

Raise $10$ to the power of $4$ .

$\frac{d}{d x} [9 x^{\frac{7}{5}} - 5 x^{2} + 10000]$

Step 1.1.2

By the Sum Rule, the derivative of $9 x^{\frac{7}{5}} - 5 x^{2} + 10000$ with respect to $x$ is $\frac{d}{d x} [9 x^{\frac{7}{5}}] + \frac{d}{d x} [- 5 x^{2}] + \frac{d}{d x} [10000]$ .

$\frac{d}{d x} [9 x^{\frac{7}{5}}] + \frac{d}{d x} [- 5 x^{2}] + \frac{d}{d x} [10000]$

Step 1.2

Evaluate $\frac{d}{d x} [9 x^{\frac{7}{5}}]$ .

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Step 1.2.1

Since $9$ is constant with respect to $x$ , the derivative of $9 x^{\frac{7}{5}}$ with respect to $x$ is $9 \frac{d}{d x} [x^{\frac{7}{5}}]$ .

$9 \frac{d}{d x} [x^{\frac{7}{5}}] + \frac{d}{d x} [- 5 x^{2}] + \frac{d}{d x} [10000]$

Step 1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{7}{5}$ .

$9 (\frac{7}{5} x^{\frac{7}{5} - 1}) + \frac{d}{d x} [- 5 x^{2}] + \frac{d}{d x} [10000]$

Step 1.2.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$9 (\frac{7}{5} x^{\frac{7}{5} - 1 \cdot \frac{5}{5}}) + \frac{d}{d x} [- 5 x^{2}] + \frac{d}{d x} [10000]$

Step 1.2.4

Combine $- 1$ and $\frac{5}{5}$ .

$9 (\frac{7}{5} x^{\frac{7}{5} + \frac{- 1 \cdot 5}{5}}) + \frac{d}{d x} [- 5 x^{2}] + \frac{d}{d x} [10000]$

Step 1.2.5

Combine the numerators over the common denominator.

$9 (\frac{7}{5} x^{\frac{7 - 1 \cdot 5}{5}}) + \frac{d}{d x} [- 5 x^{2}] + \frac{d}{d x} [10000]$

Step 1.2.6

Simplify the numerator.

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Step 1.2.6.1

Multiply $- 1$ by $5$ .

$9 (\frac{7}{5} x^{\frac{7 - 5}{5}}) + \frac{d}{d x} [- 5 x^{2}] + \frac{d}{d x} [10000]$

Step 1.2.6.2

Subtract $5$ from $7$ .

$9 (\frac{7}{5} x^{\frac{2}{5}}) + \frac{d}{d x} [- 5 x^{2}] + \frac{d}{d x} [10000]$

Step 1.2.7

Combine $\frac{7}{5}$ and $x^{\frac{2}{5}}$ .

$9 \frac{7 x^{\frac{2}{5}}}{5} + \frac{d}{d x} [- 5 x^{2}] + \frac{d}{d x} [10000]$

Step 1.2.8

Combine $9$ and $\frac{7 x^{\frac{2}{5}}}{5}$ .

$\frac{9 (7 x^{\frac{2}{5}})}{5} + \frac{d}{d x} [- 5 x^{2}] + \frac{d}{d x} [10000]$

Step 1.2.9

Multiply $7$ by $9$ .

$\frac{63 x^{\frac{2}{5}}}{5} + \frac{d}{d x} [- 5 x^{2}] + \frac{d}{d x} [10000]$

Step 1.3

Evaluate $\frac{d}{d x} [- 5 x^{2}]$ .

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Step 1.3.1

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 x^{2}$ with respect to $x$ is $- 5 \frac{d}{d x} [x^{2}]$ .

$\frac{63 x^{\frac{2}{5}}}{5} - 5 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [10000]$

Step 1.3.2

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

$\frac{63 x^{\frac{2}{5}}}{5} - 5 (2 x) + \frac{d}{d x} [10000]$

Step 1.3.3

Multiply $2$ by $- 5$ .

$\frac{63 x^{\frac{2}{5}}}{5} - 10 x + \frac{d}{d x} [10000]$

Step 1.4

Since $10000$ is constant with respect to $x$ , the derivative of $10000$ with respect to $x$ is $0$ .

$\frac{63 x^{\frac{2}{5}}}{5} - 10 x + 0$

Step 1.5

Simplify.

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Step 1.5.1

Add $\frac{63 x^{\frac{2}{5}}}{5} - 10 x$ and $0$ .

$\frac{63 x^{\frac{2}{5}}}{5} - 10 x$

Step 1.5.2

Reorder terms.

$- 10 x + \frac{63 x^{\frac{2}{5}}}{5}$

Step 2

Find the second derivative of the function.

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Step 2.1

By the Sum Rule, the derivative of $- 10 x + \frac{63 x^{\frac{2}{5}}}{5}$ with respect to $x$ is $\frac{d}{d x} [- 10 x] + \frac{d}{d x} [\frac{63 x^{\frac{2}{5}}}{5}]$ .

$f'' (x) = \frac{d}{d x} (- 10 x) + \frac{d}{d x} (\frac{63 x^{\frac{2}{5}}}{5})$

Step 2.2

Evaluate $\frac{d}{d x} [- 10 x]$ .

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Step 2.2.1

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

$f'' (x) = - 10 \frac{d}{d x} x + \frac{d}{d x} (\frac{63 x^{\frac{2}{5}}}{5})$

Step 2.2.2

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

$f'' (x) = - 10 \cdot 1 + \frac{d}{d x} (\frac{63 x^{\frac{2}{5}}}{5})$

Step 2.2.3

Multiply $- 10$ by $1$ .

$f'' (x) = - 10 + \frac{d}{d x} (\frac{63 x^{\frac{2}{5}}}{5})$

Step 2.3

Evaluate $\frac{d}{d x} [\frac{63 x^{\frac{2}{5}}}{5}]$ .

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Step 2.3.1

Since $\frac{63}{5}$ is constant with respect to $x$ , the derivative of $\frac{63 x^{\frac{2}{5}}}{5}$ with respect to $x$ is $\frac{63}{5} \frac{d}{d x} [x^{\frac{2}{5}}]$ .

$f'' (x) = - 10 + \frac{63}{5} \cdot \frac{d}{d x} (x^{\frac{2}{5}})$

Step 2.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{2}{5}$ .

$f'' (x) = - 10 + \frac{63}{5} \cdot (\frac{2}{5} x^{\frac{2}{5} - 1})$

Step 2.3.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$f'' (x) = - 10 + \frac{63}{5} \cdot (\frac{2}{5} x^{\frac{2}{5} - 1 \cdot \frac{5}{5}})$

Step 2.3.4

Combine $- 1$ and $\frac{5}{5}$ .

$f'' (x) = - 10 + \frac{63}{5} \cdot (\frac{2}{5} x^{\frac{2}{5} + \frac{- 1 \cdot 5}{5}})$

Step 2.3.5

Combine the numerators over the common denominator.

$f'' (x) = - 10 + \frac{63}{5} \cdot (\frac{2}{5} x^{\frac{2 - 1 \cdot 5}{5}})$

Step 2.3.6

Simplify the numerator.

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Step 2.3.6.1

Multiply $- 1$ by $5$ .

$f'' (x) = - 10 + \frac{63}{5} \cdot (\frac{2}{5} x^{\frac{2 - 5}{5}})$

Step 2.3.6.2

Subtract $5$ from $2$ .

$f'' (x) = - 10 + \frac{63}{5} \cdot (\frac{2}{5} x^{\frac{- 3}{5}})$

Step 2.3.7

Move the negative in front of the fraction.

$f'' (x) = - 10 + \frac{63}{5} \cdot (\frac{2}{5} x^{- \frac{3}{5}})$

Step 2.3.8

Combine $\frac{2}{5}$ and $x^{- \frac{3}{5}}$ .

$f'' (x) = - 10 + \frac{63}{5} \cdot \frac{2 x^{- \frac{3}{5}}}{5}$

Step 2.3.9

Multiply $\frac{63}{5}$ by $\frac{2 x^{- \frac{3}{5}}}{5}$ .

$f'' (x) = - 10 + \frac{63 (2 x^{- \frac{3}{5}})}{5 \cdot 5}$

Step 2.3.10

Multiply $2$ by $63$ .

$f'' (x) = - 10 + \frac{126 x^{- \frac{3}{5}}}{5 \cdot 5}$

Step 2.3.11

Multiply $5$ by $5$ .

$f'' (x) = - 10 + \frac{126 x^{- \frac{3}{5}}}{25}$

Step 2.3.12

Move $x^{- \frac{3}{5}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (x) = - 10 + \frac{126}{25 x^{\frac{3}{5}}}$

Step 3

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

$- 10 x + \frac{63 x^{\frac{2}{5}}}{5} = 0$

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 using the Sum Rule.

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Step 4.1.1.1

Raise $10$ to the power of $4$ .

$\frac{d}{d x} [9 x^{\frac{7}{5}} - 5 x^{2} + 10000]$

Step 4.1.1.2

By the Sum Rule, the derivative of $9 x^{\frac{7}{5}} - 5 x^{2} + 10000$ with respect to $x$ is $\frac{d}{d x} [9 x^{\frac{7}{5}}] + \frac{d}{d x} [- 5 x^{2}] + \frac{d}{d x} [10000]$ .

$\frac{d}{d x} [9 x^{\frac{7}{5}}] + \frac{d}{d x} [- 5 x^{2}] + \frac{d}{d x} [10000]$

Step 4.1.2

Evaluate $\frac{d}{d x} [9 x^{\frac{7}{5}}]$ .

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Step 4.1.2.1

Since $9$ is constant with respect to $x$ , the derivative of $9 x^{\frac{7}{5}}$ with respect to $x$ is $9 \frac{d}{d x} [x^{\frac{7}{5}}]$ .

$9 \frac{d}{d x} [x^{\frac{7}{5}}] + \frac{d}{d x} [- 5 x^{2}] + \frac{d}{d x} [10000]$

Step 4.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{7}{5}$ .

$9 (\frac{7}{5} x^{\frac{7}{5} - 1}) + \frac{d}{d x} [- 5 x^{2}] + \frac{d}{d x} [10000]$

Step 4.1.2.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$9 (\frac{7}{5} x^{\frac{7}{5} - 1 \cdot \frac{5}{5}}) + \frac{d}{d x} [- 5 x^{2}] + \frac{d}{d x} [10000]$

Step 4.1.2.4

Combine $- 1$ and $\frac{5}{5}$ .

$9 (\frac{7}{5} x^{\frac{7}{5} + \frac{- 1 \cdot 5}{5}}) + \frac{d}{d x} [- 5 x^{2}] + \frac{d}{d x} [10000]$

Step 4.1.2.5

Combine the numerators over the common denominator.

$9 (\frac{7}{5} x^{\frac{7 - 1 \cdot 5}{5}}) + \frac{d}{d x} [- 5 x^{2}] + \frac{d}{d x} [10000]$

Step 4.1.2.6

Simplify the numerator.

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Step 4.1.2.6.1

Multiply $- 1$ by $5$ .

$9 (\frac{7}{5} x^{\frac{7 - 5}{5}}) + \frac{d}{d x} [- 5 x^{2}] + \frac{d}{d x} [10000]$

Step 4.1.2.6.2

Subtract $5$ from $7$ .

$9 (\frac{7}{5} x^{\frac{2}{5}}) + \frac{d}{d x} [- 5 x^{2}] + \frac{d}{d x} [10000]$

Step 4.1.2.7

Combine $\frac{7}{5}$ and $x^{\frac{2}{5}}$ .

$9 \frac{7 x^{\frac{2}{5}}}{5} + \frac{d}{d x} [- 5 x^{2}] + \frac{d}{d x} [10000]$

Step 4.1.2.8

Combine $9$ and $\frac{7 x^{\frac{2}{5}}}{5}$ .

$\frac{9 (7 x^{\frac{2}{5}})}{5} + \frac{d}{d x} [- 5 x^{2}] + \frac{d}{d x} [10000]$

Step 4.1.2.9

Multiply $7$ by $9$ .

$\frac{63 x^{\frac{2}{5}}}{5} + \frac{d}{d x} [- 5 x^{2}] + \frac{d}{d x} [10000]$

Step 4.1.3

Evaluate $\frac{d}{d x} [- 5 x^{2}]$ .

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Step 4.1.3.1

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 x^{2}$ with respect to $x$ is $- 5 \frac{d}{d x} [x^{2}]$ .

$\frac{63 x^{\frac{2}{5}}}{5} - 5 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [10000]$

Step 4.1.3.2

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

$\frac{63 x^{\frac{2}{5}}}{5} - 5 (2 x) + \frac{d}{d x} [10000]$

Step 4.1.3.3

Multiply $2$ by $- 5$ .

$\frac{63 x^{\frac{2}{5}}}{5} - 10 x + \frac{d}{d x} [10000]$

Step 4.1.4

Since $10000$ is constant with respect to $x$ , the derivative of $10000$ with respect to $x$ is $0$ .

$\frac{63 x^{\frac{2}{5}}}{5} - 10 x + 0$

Step 4.1.5

Simplify.

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Step 4.1.5.1

Add $\frac{63 x^{\frac{2}{5}}}{5} - 10 x$ and $0$ .

$\frac{63 x^{\frac{2}{5}}}{5} - 10 x$

Step 4.1.5.2

Reorder terms.

$f' (x) = - 10 x + \frac{63 x^{\frac{2}{5}}}{5}$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $- 10 x + \frac{63 x^{\frac{2}{5}}}{5}$ .

$- 10 x + \frac{63 x^{\frac{2}{5}}}{5}$

Step 5

Set the first derivative equal to $0$ then solve the equation $- 10 x + \frac{63 x^{\frac{2}{5}}}{5} = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$- 10 x + \frac{63 x^{\frac{2}{5}}}{5} = 0$

Step 5.2

Find a common factor $x^{\frac{2}{5}}$ that is present in each term.

$- 10 {(x^{\frac{2}{5}})}^{\frac{5}{2}} + \frac{63 x^{\frac{2}{5}}}{5}$

Step 5.3

Substitute $u$ for $x^{\frac{2}{5}}$ .

$- 10 {(u)}^{\frac{5}{2}} + \frac{63 (u)}{5} = 0$

Step 5.4

Solve for $u$ .

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Step 5.4.1

Factor $u$ out of $- 10 u^{\frac{5}{2}} + \frac{63 u}{5}$ .

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Step 5.4.1.1

Factor $u$ out of $- 10 u^{\frac{5}{2}}$ .

$u (- 10 u^{\frac{3}{2}}) + \frac{63 u}{5} = 0$

Step 5.4.1.2

Factor $u$ out of $\frac{63 u}{5}$ .

$u (- 10 u^{\frac{3}{2}}) + u (\frac{63}{5}) = 0$

Step 5.4.1.3

Factor $u$ out of $u (- 10 u^{\frac{3}{2}}) + u \frac{63}{5}$ .

$u (- 10 u^{\frac{3}{2}} + \frac{63}{5}) = 0$

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 $0$ .

$u = 0$

$- 10 u^{\frac{3}{2}} + \frac{63}{5} = 0$

Step 5.4.3

Set $u$ equal to $0$ .

$u = 0$

Step 5.4.4

Set $- 10 u^{\frac{3}{2}} + \frac{63}{5}$ equal to $0$ and solve for $u$ .

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Step 5.4.4.1

Set $- 10 u^{\frac{3}{2}} + \frac{63}{5}$ equal to $0$ .

$- 10 u^{\frac{3}{2}} + \frac{63}{5} = 0$

Step 5.4.4.2

Solve $- 10 u^{\frac{3}{2}} + \frac{63}{5} = 0$ for $u$ .

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Step 5.4.4.2.1

Subtract $\frac{63}{5}$ from both sides of the equation.

$- 10 u^{\frac{3}{2}} = - \frac{63}{5}$

Step 5.4.4.2.2

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

${(- 10 u^{\frac{3}{2}})}^{\frac{2}{3}} = {(- \frac{63}{5})}^{\frac{2}{3}}$

Step 5.4.4.2.3

Simplify the exponent.

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Step 5.4.4.2.3.1

Simplify the left side.

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Step 5.4.4.2.3.1.1

Simplify ${(- 10 u^{\frac{3}{2}})}^{\frac{2}{3}}$ .

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Step 5.4.4.2.3.1.1.1

Apply the product rule to $- 10 u^{\frac{3}{2}}$ .

${(- 10)}^{\frac{2}{3}} {(u^{\frac{3}{2}})}^{\frac{2}{3}} = {(- \frac{63}{5})}^{\frac{2}{3}}$

Step 5.4.4.2.3.1.1.2

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

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Step 5.4.4.2.3.1.1.2.1

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

${(- 10)}^{\frac{2}{3}} u^{\frac{3}{2} \cdot \frac{2}{3}} = {(- \frac{63}{5})}^{\frac{2}{3}}$

Step 5.4.4.2.3.1.1.2.2

Cancel the common factor of $3$ .

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Step 5.4.4.2.3.1.1.2.2.1

Cancel the common factor.

${(- 10)}^{\frac{2}{3}} u^{\frac{3}{2} \cdot \frac{2}{3}} = {(- \frac{63}{5})}^{\frac{2}{3}}$

Step 5.4.4.2.3.1.1.2.2.2

Rewrite the expression.

${(- 10)}^{\frac{2}{3}} u^{\frac{1}{2} \cdot 2} = {(- \frac{63}{5})}^{\frac{2}{3}}$

Step 5.4.4.2.3.1.1.2.3

Cancel the common factor of $2$ .

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Step 5.4.4.2.3.1.1.2.3.1

Cancel the common factor.

${(- 10)}^{\frac{2}{3}} u^{\frac{1}{2} \cdot 2} = {(- \frac{63}{5})}^{\frac{2}{3}}$

Step 5.4.4.2.3.1.1.2.3.2

Rewrite the expression.

${(- 10)}^{\frac{2}{3}} u^{1} = {(- \frac{63}{5})}^{\frac{2}{3}}$

Step 5.4.4.2.3.1.1.3

Simplify.

${(- 10)}^{\frac{2}{3}} u = {(- \frac{63}{5})}^{\frac{2}{3}}$

Step 5.4.4.2.3.1.1.4

Reorder factors in ${(- 10)}^{\frac{2}{3}} u$ .

$u {(- 10)}^{\frac{2}{3}} = {(- \frac{63}{5})}^{\frac{2}{3}}$

Step 5.4.4.2.3.2

Simplify the right side.

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Step 5.4.4.2.3.2.1

Simplify ${(- \frac{63}{5})}^{\frac{2}{3}}$ .

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Step 5.4.4.2.3.2.1.1

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

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Step 5.4.4.2.3.2.1.1.1

Apply the product rule to $- \frac{63}{5}$ .

$u {(- 10)}^{\frac{2}{3}} = {(- 1)}^{\frac{2}{3}} {(\frac{63}{5})}^{\frac{2}{3}}$

Step 5.4.4.2.3.2.1.1.2

Apply the product rule to $\frac{63}{5}$ .

$u {(- 10)}^{\frac{2}{3}} = {(- 1)}^{\frac{2}{3}} \frac{63^{\frac{2}{3}}}{5^{\frac{2}{3}}}$

Step 5.4.4.2.3.2.1.2

Simplify the expression.

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Step 5.4.4.2.3.2.1.2.1

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$u {(- 10)}^{\frac{2}{3}} = {({(- 1)}^{3})}^{\frac{2}{3}} \frac{63^{\frac{2}{3}}}{5^{\frac{2}{3}}}$

Step 5.4.4.2.3.2.1.2.2

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

$u {(- 10)}^{\frac{2}{3}} = {(- 1)}^{3 (\frac{2}{3})} \frac{63^{\frac{2}{3}}}{5^{\frac{2}{3}}}$

Step 5.4.4.2.3.2.1.3

Cancel the common factor of $3$ .

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Step 5.4.4.2.3.2.1.3.1

Cancel the common factor.

$u {(- 10)}^{\frac{2}{3}} = {(- 1)}^{3 (\frac{2}{3})} \frac{63^{\frac{2}{3}}}{5^{\frac{2}{3}}}$

Step 5.4.4.2.3.2.1.3.2

Rewrite the expression.

$u {(- 10)}^{\frac{2}{3}} = {(- 1)}^{2} \frac{63^{\frac{2}{3}}}{5^{\frac{2}{3}}}$

Step 5.4.4.2.3.2.1.4

Simplify the expression.

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Step 5.4.4.2.3.2.1.4.1

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

$u {(- 10)}^{\frac{2}{3}} = 1 \frac{63^{\frac{2}{3}}}{5^{\frac{2}{3}}}$

Step 5.4.4.2.3.2.1.4.2

Multiply $\frac{63^{\frac{2}{3}}}{5^{\frac{2}{3}}}$ by $1$ .

$u {(- 10)}^{\frac{2}{3}} = \frac{63^{\frac{2}{3}}}{5^{\frac{2}{3}}}$

Step 5.4.4.2.4

Divide each term in $u {(- 10)}^{\frac{2}{3}} = \frac{63^{\frac{2}{3}}}{5^{\frac{2}{3}}}$ by ${(- 10)}^{\frac{2}{3}}$ and simplify.

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Step 5.4.4.2.4.1

Divide each term in $u {(- 10)}^{\frac{2}{3}} = \frac{63^{\frac{2}{3}}}{5^{\frac{2}{3}}}$ by ${(- 10)}^{\frac{2}{3}}$ .

$\frac{u {(- 10)}^{\frac{2}{3}}}{{(- 10)}^{\frac{2}{3}}} = \frac{\frac{63^{\frac{2}{3}}}{5^{\frac{2}{3}}}}{{(- 10)}^{\frac{2}{3}}}$

Step 5.4.4.2.4.2

Simplify the left side.

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Step 5.4.4.2.4.2.1

Cancel the common factor.

$\frac{u {(- 10)}^{\frac{2}{3}}}{{(- 10)}^{\frac{2}{3}}} = \frac{\frac{63^{\frac{2}{3}}}{5^{\frac{2}{3}}}}{{(- 10)}^{\frac{2}{3}}}$

Step 5.4.4.2.4.2.2

Divide $u$ by $1$ .

$u = \frac{\frac{63^{\frac{2}{3}}}{5^{\frac{2}{3}}}}{{(- 10)}^{\frac{2}{3}}}$

Step 5.4.4.2.4.3

Simplify the right side.

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Step 5.4.4.2.4.3.1

Multiply the numerator by the reciprocal of the denominator.

$u = \frac{63^{\frac{2}{3}}}{5^{\frac{2}{3}}} \cdot \frac{1}{{(- 10)}^{\frac{2}{3}}}$

Step 5.4.4.2.4.3.2

Combine.

$u = \frac{63^{\frac{2}{3}} \cdot 1}{5^{\frac{2}{3}} {(- 10)}^{\frac{2}{3}}}$

Step 5.4.4.2.4.3.3

Multiply $63^{\frac{2}{3}}$ by $1$ .

$u = \frac{63^{\frac{2}{3}}}{5^{\frac{2}{3}} {(- 10)}^{\frac{2}{3}}}$

Step 5.4.5

The final solution is all the values that make $u (- 10 u^{\frac{3}{2}} + \frac{63}{5}) = 0$ true.

$u = 0, \frac{63^{\frac{2}{3}}}{5^{\frac{2}{3}} {(- 10)}^{\frac{2}{3}}}$

Step 5.5

Substitute $x$ for $u$ .

$x^{\frac{2}{5}} = 0, \frac{63^{\frac{2}{3}}}{5^{\frac{2}{3}} {(- 10)}^{\frac{2}{3}}}$

Step 5.6

Solve for $x^{\frac{2}{5}} = 0$ for $x$ .

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Step 5.6.1

Raise each side of the equation to the power of $\frac{5}{2}$ to eliminate the fractional exponent on the left side.

${(x^{\frac{2}{5}})}^{\frac{5}{2}} = \pm 0^{\frac{5}{2}}$

Step 5.6.2

Simplify the exponent.

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Step 5.6.2.1

Simplify the left side.

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Step 5.6.2.1.1

Simplify ${(x^{\frac{2}{5}})}^{\frac{5}{2}}$ .

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Step 5.6.2.1.1.1

Multiply the exponents in ${(x^{\frac{2}{5}})}^{\frac{5}{2}}$ .

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Step 5.6.2.1.1.1.1

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

$x^{\frac{2}{5} \cdot \frac{5}{2}} = \pm 0^{\frac{5}{2}}$

Step 5.6.2.1.1.1.2

Cancel the common factor of $2$ .

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Step 5.6.2.1.1.1.2.1

Cancel the common factor.

$x^{\frac{2}{5} \cdot \frac{5}{2}} = \pm 0^{\frac{5}{2}}$

Step 5.6.2.1.1.1.2.2

Rewrite the expression.

$x^{\frac{1}{5} \cdot 5} = \pm 0^{\frac{5}{2}}$

Step 5.6.2.1.1.1.3

Cancel the common factor of $5$ .

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Step 5.6.2.1.1.1.3.1

Cancel the common factor.

$x^{\frac{1}{5} \cdot 5} = \pm 0^{\frac{5}{2}}$

Step 5.6.2.1.1.1.3.2

Rewrite the expression.

$x^{1} = \pm 0^{\frac{5}{2}}$

Step 5.6.2.1.1.2

Simplify.

$x = \pm 0^{\frac{5}{2}}$

Step 5.6.2.2

Simplify the right side.

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Step 5.6.2.2.1

Simplify $\pm 0^{\frac{5}{2}}$ .

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Step 5.6.2.2.1.1

Simplify the expression.

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Step 5.6.2.2.1.1.1

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

$x = \pm {(0^{2})}^{\frac{5}{2}}$

Step 5.6.2.2.1.1.2

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

$x = \pm 0^{2 (\frac{5}{2})}$

Step 5.6.2.2.1.2

Cancel the common factor of $2$ .

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Step 5.6.2.2.1.2.1

Cancel the common factor.

$x = \pm 0^{2 (\frac{5}{2})}$

Step 5.6.2.2.1.2.2

Rewrite the expression.

$x = \pm 0^{5}$

Step 5.6.2.2.1.3

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

$x = \pm 0$

Step 5.6.2.2.1.4

Plus or minus $0$ is $0$ .

$x = 0$

Step 5.7

Solve for $x^{\frac{2}{5}} = \frac{63^{\frac{2}{3}}}{5^{\frac{2}{3}} {(- 10)}^{\frac{2}{3}}}$ for $x$ .

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Step 5.7.1

Raise each side of the equation to the power of $\frac{5}{2}$ to eliminate the fractional exponent on the left side.

${(x^{\frac{2}{5}})}^{\frac{5}{2}} = \pm {(\frac{63^{\frac{2}{3}}}{5^{\frac{2}{3}} {(- 10)}^{\frac{2}{3}}})}^{\frac{5}{2}}$

Step 5.7.2

Simplify the exponent.

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Step 5.7.2.1

Simplify the left side.

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Step 5.7.2.1.1

Simplify ${(x^{\frac{2}{5}})}^{\frac{5}{2}}$ .

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Step 5.7.2.1.1.1

Multiply the exponents in ${(x^{\frac{2}{5}})}^{\frac{5}{2}}$ .

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Step 5.7.2.1.1.1.1

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

$x^{\frac{2}{5} \cdot \frac{5}{2}} = \pm {(\frac{63^{\frac{2}{3}}}{5^{\frac{2}{3}} {(- 10)}^{\frac{2}{3}}})}^{\frac{5}{2}}$

Step 5.7.2.1.1.1.2

Cancel the common factor of $2$ .

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Step 5.7.2.1.1.1.2.1

Cancel the common factor.

$x^{\frac{2}{5} \cdot \frac{5}{2}} = \pm {(\frac{63^{\frac{2}{3}}}{5^{\frac{2}{3}} {(- 10)}^{\frac{2}{3}}})}^{\frac{5}{2}}$

Step 5.7.2.1.1.1.2.2

Rewrite the expression.

$x^{\frac{1}{5} \cdot 5} = \pm {(\frac{63^{\frac{2}{3}}}{5^{\frac{2}{3}} {(- 10)}^{\frac{2}{3}}})}^{\frac{5}{2}}$

Step 5.7.2.1.1.1.3

Cancel the common factor of $5$ .

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Step 5.7.2.1.1.1.3.1

Cancel the common factor.

$x^{\frac{1}{5} \cdot 5} = \pm {(\frac{63^{\frac{2}{3}}}{5^{\frac{2}{3}} {(- 10)}^{\frac{2}{3}}})}^{\frac{5}{2}}$

Step 5.7.2.1.1.1.3.2

Rewrite the expression.

$x^{1} = \pm {(\frac{63^{\frac{2}{3}}}{5^{\frac{2}{3}} {(- 10)}^{\frac{2}{3}}})}^{\frac{5}{2}}$

Step 5.7.2.1.1.2

Simplify.

$x = \pm {(\frac{63^{\frac{2}{3}}}{5^{\frac{2}{3}} {(- 10)}^{\frac{2}{3}}})}^{\frac{5}{2}}$

Step 5.7.2.2

Simplify the right side.

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Step 5.7.2.2.1

Simplify $\pm {(\frac{63^{\frac{2}{3}}}{5^{\frac{2}{3}} {(- 10)}^{\frac{2}{3}}})}^{\frac{5}{2}}$ .

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Step 5.7.2.2.1.1

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

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Step 5.7.2.2.1.1.1

Apply the product rule to $\frac{63^{\frac{2}{3}}}{5^{\frac{2}{3}} {(- 10)}^{\frac{2}{3}}}$ .

$x = \pm \frac{{(63^{\frac{2}{3}})}^{\frac{5}{2}}}{{(5^{\frac{2}{3}} {(- 10)}^{\frac{2}{3}})}^{\frac{5}{2}}}$

Step 5.7.2.2.1.1.2

Apply the product rule to $5^{\frac{2}{3}} {(- 10)}^{\frac{2}{3}}$ .

$x = \pm \frac{{(63^{\frac{2}{3}})}^{\frac{5}{2}}}{{(5^{\frac{2}{3}})}^{\frac{5}{2}} {({(- 10)}^{\frac{2}{3}})}^{\frac{5}{2}}}$

Step 5.7.2.2.1.2

Multiply the exponents in ${(63^{\frac{2}{3}})}^{\frac{5}{2}}$ .

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Step 5.7.2.2.1.2.1

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

$x = \pm \frac{63^{\frac{2}{3} \cdot \frac{5}{2}}}{{(5^{\frac{2}{3}})}^{\frac{5}{2}} {({(- 10)}^{\frac{2}{3}})}^{\frac{5}{2}}}$

Step 5.7.2.2.1.2.2

Cancel the common factor of $2$ .

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Step 5.7.2.2.1.2.2.1

Cancel the common factor.

$x = \pm \frac{63^{\frac{2}{3} \cdot \frac{5}{2}}}{{(5^{\frac{2}{3}})}^{\frac{5}{2}} {({(- 10)}^{\frac{2}{3}})}^{\frac{5}{2}}}$

Step 5.7.2.2.1.2.2.2

Rewrite the expression.

$x = \pm \frac{63^{\frac{1}{3} \cdot 5}}{{(5^{\frac{2}{3}})}^{\frac{5}{2}} {({(- 10)}^{\frac{2}{3}})}^{\frac{5}{2}}}$

Step 5.7.2.2.1.2.3

Combine $\frac{1}{3}$ and $5$ .

$x = \pm \frac{63^{\frac{5}{3}}}{{(5^{\frac{2}{3}})}^{\frac{5}{2}} {({(- 10)}^{\frac{2}{3}})}^{\frac{5}{2}}}$

Step 5.7.2.2.1.3

Simplify the denominator.

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Step 5.7.2.2.1.3.1

Multiply the exponents in ${(5^{\frac{2}{3}})}^{\frac{5}{2}}$ .

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Step 5.7.2.2.1.3.1.1

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

$x = \pm \frac{63^{\frac{5}{3}}}{5^{\frac{2}{3} \cdot \frac{5}{2}} {({(- 10)}^{\frac{2}{3}})}^{\frac{5}{2}}}$

Step 5.7.2.2.1.3.1.2

Cancel the common factor of $2$ .

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Step 5.7.2.2.1.3.1.2.1

Cancel the common factor.

$x = \pm \frac{63^{\frac{5}{3}}}{5^{\frac{2}{3} \cdot \frac{5}{2}} {({(- 10)}^{\frac{2}{3}})}^{\frac{5}{2}}}$

Step 5.7.2.2.1.3.1.2.2

Rewrite the expression.

$x = \pm \frac{63^{\frac{5}{3}}}{5^{\frac{1}{3} \cdot 5} {({(- 10)}^{\frac{2}{3}})}^{\frac{5}{2}}}$

Step 5.7.2.2.1.3.1.3

Combine $\frac{1}{3}$ and $5$ .

$x = \pm \frac{63^{\frac{5}{3}}}{5^{\frac{5}{3}} {({(- 10)}^{\frac{2}{3}})}^{\frac{5}{2}}}$

Step 5.7.2.2.1.3.2

Multiply the exponents in ${({(- 10)}^{\frac{2}{3}})}^{\frac{5}{2}}$ .

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Step 5.7.2.2.1.3.2.1

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

$x = \pm \frac{63^{\frac{5}{3}}}{5^{\frac{5}{3}} {(- 10)}^{\frac{2}{3} \cdot \frac{5}{2}}}$

Step 5.7.2.2.1.3.2.2

Cancel the common factor of $2$ .

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Step 5.7.2.2.1.3.2.2.1

Cancel the common factor.

$x = \pm \frac{63^{\frac{5}{3}}}{5^{\frac{5}{3}} {(- 10)}^{\frac{2}{3} \cdot \frac{5}{2}}}$

Step 5.7.2.2.1.3.2.2.2

Rewrite the expression.

$x = \pm \frac{63^{\frac{5}{3}}}{5^{\frac{5}{3}} {(- 10)}^{\frac{1}{3} \cdot 5}}$

Step 5.7.2.2.1.3.2.3

Combine $\frac{1}{3}$ and $5$ .

$x = \pm \frac{63^{\frac{5}{3}}}{5^{\frac{5}{3}} {(- 10)}^{\frac{5}{3}}}$

Step 5.7.3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 5.7.3.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \frac{63^{\frac{5}{3}}}{5^{\frac{5}{3}} {(- 10)}^{\frac{5}{3}}}$

Step 5.7.3.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \frac{63^{\frac{5}{3}}}{5^{\frac{5}{3}} {(- 10)}^{\frac{5}{3}}}$

Step 5.7.3.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{63^{\frac{5}{3}}}{5^{\frac{5}{3}} {(- 10)}^{\frac{5}{3}}}, - \frac{63^{\frac{5}{3}}}{5^{\frac{5}{3}} {(- 10)}^{\frac{5}{3}}}$

Step 5.8

List all of the solutions.

$x = 0, \frac{63^{\frac{5}{3}}}{5^{\frac{5}{3}} {(- 10)}^{\frac{5}{3}}}, - \frac{63^{\frac{5}{3}}}{5^{\frac{5}{3}} {(- 10)}^{\frac{5}{3}}}$

Step 5.9

Exclude the solutions that do not make $- 10 x + \frac{63 x^{\frac{2}{5}}}{5} = 0$ true.

$x = 0, - \frac{63^{\frac{5}{3}}}{5^{\frac{5}{3}} {(- 10)}^{\frac{5}{3}}}$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 7

Critical points to evaluate.

$x = 0, - \frac{63^{\frac{5}{3}}}{5^{\frac{5}{3}} {(- 10)}^{\frac{5}{3}}}$

Step 8

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.

$- 10 + \frac{126}{25 {(0)}^{\frac{3}{5}}}$

Step 9

Evaluate the second derivative.

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Step 9.1

Simplify the expression.

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Step 9.1.1

Rewrite $0$ as $0^{5}$ .

$- 10 + \frac{126}{25 {(0^{5})}^{\frac{3}{5}}}$

Step 9.1.2

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

$- 10 + \frac{126}{25 \cdot 0^{5 (\frac{3}{5})}}$

Step 9.2

Cancel the common factor of $5$ .

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Step 9.2.1

Cancel the common factor.

$- 10 + \frac{126}{25 \cdot 0^{5 (\frac{3}{5})}}$

Step 9.2.2

Rewrite the expression.

$- 10 + \frac{126}{25 \cdot 0^{3}}$

Step 9.3

Simplify the expression.

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Step 9.3.1

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

$- 10 + \frac{126}{25 \cdot 0}$

Step 9.3.2

Multiply $25$ by $0$ .

$- 10 + \frac{126}{0}$

Step 9.3.3

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

Undefined

$- 10 + \frac{126}{0}$

Step 9.4

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

Undefined

Step 10

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

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Step 10.1

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

$(- \infty, 0) \cup (0, - \frac{63^{\frac{5}{3}}}{5^{\frac{5}{3}} {(- 10)}^{\frac{5}{3}}}) \cup (- \frac{63^{\frac{5}{3}}}{5^{\frac{5}{3}} {(- 10)}^{\frac{5}{3}}}, \infty)$

Step 10.2

Substitute any number, such as $- 2$ , from the interval $(- \infty, 0)$ in the first derivative $- 10 x + \frac{63 x^{\frac{2}{5}}}{5}$ to check if the result is negative or positive.

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Step 10.2.1

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

$f' (- 2) = - 10 \cdot - 2 + \frac{63 {(- 2)}^{\frac{2}{5}}}{5}$

Step 10.2.2

Simplify the result.

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Step 10.2.2.1

Multiply $- 10$ by $- 2$ .

$f' (- 2) = 20 + \frac{63 {(- 2)}^{\frac{2}{5}}}{5}$

Step 10.2.2.2

The final answer is $20 + \frac{63 {(- 2)}^{\frac{2}{5}}}{5}$ .

$20 + \frac{63 {(- 2)}^{\frac{2}{5}}}{5}$

Step 10.3

No local maxima or minima found for $f (x) = 9 x^{\frac{7}{5}} - 5 x^{2} + 10^{4}$ .

No local maxima or minima

Step 11