Calculus Examples

$f (x) = 2 x^{\frac{5}{2}} - 135 x - 1$

Step 1

Find the first derivative of the function.

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Step 1.1

By the Sum Rule, the derivative of $2 x^{\frac{5}{2}} - 135 x - 1$ with respect to $x$ is $\frac{d}{d x} [2 x^{\frac{5}{2}}] + \frac{d}{d x} [- 135 x] + \frac{d}{d x} [- 1]$ .

$\frac{d}{d x} [2 x^{\frac{5}{2}}] + \frac{d}{d x} [- 135 x] + \frac{d}{d x} [- 1]$

Step 1.2

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

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Step 1.2.1

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

$2 \frac{d}{d x} [x^{\frac{5}{2}}] + \frac{d}{d x} [- 135 x] + \frac{d}{d x} [- 1]$

Step 1.2.2

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

$2 (\frac{5}{2} x^{\frac{5}{2} - 1}) + \frac{d}{d x} [- 135 x] + \frac{d}{d x} [- 1]$

Step 1.2.3

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

$2 (\frac{5}{2} x^{\frac{5}{2} - 1 \cdot \frac{2}{2}}) + \frac{d}{d x} [- 135 x] + \frac{d}{d x} [- 1]$

Step 1.2.4

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

$2 (\frac{5}{2} x^{\frac{5}{2} + \frac{- 1 \cdot 2}{2}}) + \frac{d}{d x} [- 135 x] + \frac{d}{d x} [- 1]$

Step 1.2.5

Combine the numerators over the common denominator.

$2 (\frac{5}{2} x^{\frac{5 - 1 \cdot 2}{2}}) + \frac{d}{d x} [- 135 x] + \frac{d}{d x} [- 1]$

Step 1.2.6

Simplify the numerator.

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Step 1.2.6.1

Multiply $- 1$ by $2$ .

$2 (\frac{5}{2} x^{\frac{5 - 2}{2}}) + \frac{d}{d x} [- 135 x] + \frac{d}{d x} [- 1]$

Step 1.2.6.2

Subtract $2$ from $5$ .

$2 (\frac{5}{2} x^{\frac{3}{2}}) + \frac{d}{d x} [- 135 x] + \frac{d}{d x} [- 1]$

Step 1.2.7

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

$2 \frac{5 x^{\frac{3}{2}}}{2} + \frac{d}{d x} [- 135 x] + \frac{d}{d x} [- 1]$

Step 1.2.8

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

$\frac{2 (5 x^{\frac{3}{2}})}{2} + \frac{d}{d x} [- 135 x] + \frac{d}{d x} [- 1]$

Step 1.2.9

Multiply $5$ by $2$ .

$\frac{10 x^{\frac{3}{2}}}{2} + \frac{d}{d x} [- 135 x] + \frac{d}{d x} [- 1]$

Step 1.2.10

Factor $2$ out of $10 x^{\frac{3}{2}}$ .

$\frac{2 (5 x^{\frac{3}{2}})}{2} + \frac{d}{d x} [- 135 x] + \frac{d}{d x} [- 1]$

Step 1.2.11

Cancel the common factors.

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Step 1.2.11.1

Factor $2$ out of $2$ .

$\frac{2 (5 x^{\frac{3}{2}})}{2 (1)} + \frac{d}{d x} [- 135 x] + \frac{d}{d x} [- 1]$

Step 1.2.11.2

Cancel the common factor.

$\frac{2 (5 x^{\frac{3}{2}})}{2 \cdot 1} + \frac{d}{d x} [- 135 x] + \frac{d}{d x} [- 1]$

Step 1.2.11.3

Rewrite the expression.

$\frac{5 x^{\frac{3}{2}}}{1} + \frac{d}{d x} [- 135 x] + \frac{d}{d x} [- 1]$

Step 1.2.11.4

Divide $5 x^{\frac{3}{2}}$ by $1$ .

$5 x^{\frac{3}{2}} + \frac{d}{d x} [- 135 x] + \frac{d}{d x} [- 1]$

Step 1.3

Evaluate $\frac{d}{d x} [- 135 x]$ .

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Step 1.3.1

Since $- 135$ is constant with respect to $x$ , the derivative of $- 135 x$ with respect to $x$ is $- 135 \frac{d}{d x} [x]$ .

$5 x^{\frac{3}{2}} - 135 \frac{d}{d x} [x] + \frac{d}{d x} [- 1]$

Step 1.3.2

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

$5 x^{\frac{3}{2}} - 135 \cdot 1 + \frac{d}{d x} [- 1]$

Step 1.3.3

Multiply $- 135$ by $1$ .

$5 x^{\frac{3}{2}} - 135 + \frac{d}{d x} [- 1]$

Step 1.4

Differentiate using the Constant Rule.

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Step 1.4.1

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

$5 x^{\frac{3}{2}} - 135 + 0$

Step 1.4.2

Add $5 x^{\frac{3}{2}} - 135$ and $0$ .

$5 x^{\frac{3}{2}} - 135$

Step 2

Find the second derivative of the function.

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Step 2.1

By the Sum Rule, the derivative of $5 x^{\frac{3}{2}} - 135$ with respect to $x$ is $\frac{d}{d x} [5 x^{\frac{3}{2}}] + \frac{d}{d x} [- 135]$ .

$f'' (x) = \frac{d}{d x} (5 x^{\frac{3}{2}}) + \frac{d}{d x} (- 135)$

Step 2.2

Evaluate $\frac{d}{d x} [5 x^{\frac{3}{2}}]$ .

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Step 2.2.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{\frac{3}{2}}$ with respect to $x$ is $5 \frac{d}{d x} [x^{\frac{3}{2}}]$ .

$f'' (x) = 5 \frac{d}{d x} (x^{\frac{3}{2}}) + \frac{d}{d x} (- 135)$

Step 2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{3}{2}$ .

$f'' (x) = 5 (\frac{3}{2} x^{\frac{3}{2} - 1}) + \frac{d}{d x} (- 135)$

Step 2.2.3

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

$f'' (x) = 5 (\frac{3}{2} x^{\frac{3}{2} - 1 \cdot \frac{2}{2}}) + \frac{d}{d x} (- 135)$

Step 2.2.4

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

$f'' (x) = 5 (\frac{3}{2} x^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}}) + \frac{d}{d x} (- 135)$

Step 2.2.5

Combine the numerators over the common denominator.

$f'' (x) = 5 (\frac{3}{2} x^{\frac{3 - 1 \cdot 2}{2}}) + \frac{d}{d x} (- 135)$

Step 2.2.6

Simplify the numerator.

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Step 2.2.6.1

Multiply $- 1$ by $2$ .

$f'' (x) = 5 (\frac{3}{2} x^{\frac{3 - 2}{2}}) + \frac{d}{d x} (- 135)$

Step 2.2.6.2

Subtract $2$ from $3$ .

$f'' (x) = 5 (\frac{3}{2} x^{\frac{1}{2}}) + \frac{d}{d x} (- 135)$

Step 2.2.7

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

$f'' (x) = 5 (\frac{3 x^{\frac{1}{2}}}{2}) + \frac{d}{d x} (- 135)$

Step 2.2.8

Combine $5$ and $\frac{3 x^{\frac{1}{2}}}{2}$ .

$f'' (x) = \frac{5 (3 x^{\frac{1}{2}})}{2} + \frac{d}{d x} (- 135)$

Step 2.2.9

Multiply $3$ by $5$ .

$f'' (x) = \frac{15 x^{\frac{1}{2}}}{2} + \frac{d}{d x} (- 135)$

Step 2.3

Differentiate using the Constant Rule.

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Step 2.3.1

Since $- 135$ is constant with respect to $x$ , the derivative of $- 135$ with respect to $x$ is $0$ .

$f'' (x) = \frac{15 x^{\frac{1}{2}}}{2} + 0$

Step 2.3.2

Add $\frac{15 x^{\frac{1}{2}}}{2}$ and $0$ .

$f'' (x) = \frac{15 x^{\frac{1}{2}}}{2}$

Step 3

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

$5 x^{\frac{3}{2}} - 135 = 0$

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

By the Sum Rule, the derivative of $2 x^{\frac{5}{2}} - 135 x - 1$ with respect to $x$ is $\frac{d}{d x} [2 x^{\frac{5}{2}}] + \frac{d}{d x} [- 135 x] + \frac{d}{d x} [- 1]$ .

$\frac{d}{d x} [2 x^{\frac{5}{2}}] + \frac{d}{d x} [- 135 x] + \frac{d}{d x} [- 1]$

Step 4.1.2

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

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Step 4.1.2.1

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

$2 \frac{d}{d x} [x^{\frac{5}{2}}] + \frac{d}{d x} [- 135 x] + \frac{d}{d x} [- 1]$

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 $n = \frac{5}{2}$ .

$2 (\frac{5}{2} x^{\frac{5}{2} - 1}) + \frac{d}{d x} [- 135 x] + \frac{d}{d x} [- 1]$

Step 4.1.2.3

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

$2 (\frac{5}{2} x^{\frac{5}{2} - 1 \cdot \frac{2}{2}}) + \frac{d}{d x} [- 135 x] + \frac{d}{d x} [- 1]$

Step 4.1.2.4

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

$2 (\frac{5}{2} x^{\frac{5}{2} + \frac{- 1 \cdot 2}{2}}) + \frac{d}{d x} [- 135 x] + \frac{d}{d x} [- 1]$

Step 4.1.2.5

Combine the numerators over the common denominator.

$2 (\frac{5}{2} x^{\frac{5 - 1 \cdot 2}{2}}) + \frac{d}{d x} [- 135 x] + \frac{d}{d x} [- 1]$

Step 4.1.2.6

Simplify the numerator.

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Step 4.1.2.6.1

Multiply $- 1$ by $2$ .

$2 (\frac{5}{2} x^{\frac{5 - 2}{2}}) + \frac{d}{d x} [- 135 x] + \frac{d}{d x} [- 1]$

Step 4.1.2.6.2

Subtract $2$ from $5$ .

$2 (\frac{5}{2} x^{\frac{3}{2}}) + \frac{d}{d x} [- 135 x] + \frac{d}{d x} [- 1]$

Step 4.1.2.7

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

$2 \frac{5 x^{\frac{3}{2}}}{2} + \frac{d}{d x} [- 135 x] + \frac{d}{d x} [- 1]$

Step 4.1.2.8

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

$\frac{2 (5 x^{\frac{3}{2}})}{2} + \frac{d}{d x} [- 135 x] + \frac{d}{d x} [- 1]$

Step 4.1.2.9

Multiply $5$ by $2$ .

$\frac{10 x^{\frac{3}{2}}}{2} + \frac{d}{d x} [- 135 x] + \frac{d}{d x} [- 1]$

Step 4.1.2.10

Factor $2$ out of $10 x^{\frac{3}{2}}$ .

$\frac{2 (5 x^{\frac{3}{2}})}{2} + \frac{d}{d x} [- 135 x] + \frac{d}{d x} [- 1]$

Step 4.1.2.11

Cancel the common factors.

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Step 4.1.2.11.1

Factor $2$ out of $2$ .

$\frac{2 (5 x^{\frac{3}{2}})}{2 (1)} + \frac{d}{d x} [- 135 x] + \frac{d}{d x} [- 1]$

Step 4.1.2.11.2

Cancel the common factor.

$\frac{2 (5 x^{\frac{3}{2}})}{2 \cdot 1} + \frac{d}{d x} [- 135 x] + \frac{d}{d x} [- 1]$

Step 4.1.2.11.3

Rewrite the expression.

$\frac{5 x^{\frac{3}{2}}}{1} + \frac{d}{d x} [- 135 x] + \frac{d}{d x} [- 1]$

Step 4.1.2.11.4

Divide $5 x^{\frac{3}{2}}$ by $1$ .

$5 x^{\frac{3}{2}} + \frac{d}{d x} [- 135 x] + \frac{d}{d x} [- 1]$

Step 4.1.3

Evaluate $\frac{d}{d x} [- 135 x]$ .

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Step 4.1.3.1

Since $- 135$ is constant with respect to $x$ , the derivative of $- 135 x$ with respect to $x$ is $- 135 \frac{d}{d x} [x]$ .

$5 x^{\frac{3}{2}} - 135 \frac{d}{d x} [x] + \frac{d}{d x} [- 1]$

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 $n = 1$ .

$5 x^{\frac{3}{2}} - 135 \cdot 1 + \frac{d}{d x} [- 1]$

Step 4.1.3.3

Multiply $- 135$ by $1$ .

$5 x^{\frac{3}{2}} - 135 + \frac{d}{d x} [- 1]$

Step 4.1.4

Differentiate using the Constant Rule.

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Step 4.1.4.1

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

$5 x^{\frac{3}{2}} - 135 + 0$

Step 4.1.4.2

Add $5 x^{\frac{3}{2}} - 135$ and $0$ .

$f' (x) = 5 x^{\frac{3}{2}} - 135$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $5 x^{\frac{3}{2}} - 135$ .

$5 x^{\frac{3}{2}} - 135$

Step 5

Set the first derivative equal to $0$ then solve the equation $5 x^{\frac{3}{2}} - 135 = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$5 x^{\frac{3}{2}} - 135 = 0$

Step 5.2

Add $135$ to both sides of the equation.

$5 x^{\frac{3}{2}} = 135$

Step 5.3

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

${(5 x^{\frac{3}{2}})}^{\frac{2}{3}} = 135^{\frac{2}{3}}$

Step 5.4

Simplify the left side.

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Step 5.4.1

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

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Step 5.4.1.1

Apply the product rule to $5 x^{\frac{3}{2}}$ .

$5^{\frac{2}{3}} {(x^{\frac{3}{2}})}^{\frac{2}{3}} = 135^{\frac{2}{3}}$

Step 5.4.1.2

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

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Step 5.4.1.2.1

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

$5^{\frac{2}{3}} x^{\frac{3}{2} \cdot \frac{2}{3}} = 135^{\frac{2}{3}}$

Step 5.4.1.2.2

Cancel the common factor of $3$ .

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Step 5.4.1.2.2.1

Cancel the common factor.

$5^{\frac{2}{3}} x^{\frac{3}{2} \cdot \frac{2}{3}} = 135^{\frac{2}{3}}$

Step 5.4.1.2.2.2

Rewrite the expression.

$5^{\frac{2}{3}} x^{\frac{1}{2} \cdot 2} = 135^{\frac{2}{3}}$

Step 5.4.1.2.3

Cancel the common factor of $2$ .

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Step 5.4.1.2.3.1

Cancel the common factor.

$5^{\frac{2}{3}} x^{\frac{1}{2} \cdot 2} = 135^{\frac{2}{3}}$

Step 5.4.1.2.3.2

Rewrite the expression.

$5^{\frac{2}{3}} x^{1} = 135^{\frac{2}{3}}$

Step 5.4.1.3

Simplify.

$5^{\frac{2}{3}} x = 135^{\frac{2}{3}}$

Step 5.4.1.4

Reorder factors in $5^{\frac{2}{3}} x$ .

$x \cdot 5^{\frac{2}{3}} = 135^{\frac{2}{3}}$

Step 5.5

Divide each term in $x \cdot 5^{\frac{2}{3}} = 135^{\frac{2}{3}}$ by $5^{\frac{2}{3}}$ and simplify.

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Step 5.5.1

Divide each term in $x \cdot 5^{\frac{2}{3}} = 135^{\frac{2}{3}}$ by $5^{\frac{2}{3}}$ .

$\frac{x \cdot 5^{\frac{2}{3}}}{5^{\frac{2}{3}}} = \frac{135^{\frac{2}{3}}}{5^{\frac{2}{3}}}$

Step 5.5.2

Simplify the left side.

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Step 5.5.2.1

Cancel the common factor.

$\frac{x \cdot 5^{\frac{2}{3}}}{5^{\frac{2}{3}}} = \frac{135^{\frac{2}{3}}}{5^{\frac{2}{3}}}$

Step 5.5.2.2

Divide $x$ by $1$ .

$x = \frac{135^{\frac{2}{3}}}{5^{\frac{2}{3}}}$

Step 5.5.3

Simplify the right side.

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Step 5.5.3.1

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

$x = {(\frac{135}{5})}^{\frac{2}{3}}$

Step 5.5.3.2

Simplify the expression.

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Step 5.5.3.2.1

Divide $135$ by $5$ .

$x = 27^{\frac{2}{3}}$

Step 5.5.3.2.2

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

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

Step 5.5.3.2.3

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

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

Step 5.5.3.3

Cancel the common factor of $3$ .

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Step 5.5.3.3.1

Cancel the common factor.

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

Step 5.5.3.3.2

Rewrite the expression.

$x = 3^{2}$

Step 5.5.3.4

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

$x = 9$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

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

$5 \sqrt{x^{3}} - 135$

Step 6.2

Set the radicand in $\sqrt{x^{3}}$ less than $0$ to find where the expression is undefined.

$x^{3} < 0$

Step 6.3

Solve for $x$ .

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Step 6.3.1

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

$\sqrt[3]{x^{3}} < \sqrt[3]{0}$

Step 6.3.2

Simplify the equation.

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Step 6.3.2.1

Simplify the left side.

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Step 6.3.2.1.1

Pull terms out from under the radical.

$x < \sqrt[3]{0}$

Step 6.3.2.2

Simplify the right side.

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Step 6.3.2.2.1

Simplify $\sqrt[3]{0}$ .

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Step 6.3.2.2.1.1

Rewrite $0$ as $0^{3}$ .

$x < \sqrt[3]{0^{3}}$

Step 6.3.2.2.1.2

Pull terms out from under the radical.

$x < 0$

Step 6.4

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 $0$ .

$x < 0$

$(- \infty, 0)$

$x < 0$

$(- \infty, 0)$

Step 7

Critical points to evaluate.

$x = 9$

Step 8

Evaluate the second derivative at $x = 9$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{15 {(9)}^{\frac{1}{2}}}{2}$

Step 9

Evaluate the second derivative.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Rewrite $9$ as $3^{2}$ .

$\frac{15 \cdot {(3^{2})}^{\frac{1}{2}}}{2}$

Step 9.1.2

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

$\frac{15 \cdot 3^{2 (\frac{1}{2})}}{2}$

Step 9.1.3

Cancel the common factor of $2$ .

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Step 9.1.3.1

Cancel the common factor.

$\frac{15 \cdot 3^{2 (\frac{1}{2})}}{2}$

Step 9.1.3.2

Rewrite the expression.

$\frac{15 \cdot 3^{1}}{2}$

Step 9.1.4

Evaluate the exponent.

$\frac{15 \cdot 3}{2}$

Step 9.2

Multiply $15$ by $3$ .

$\frac{45}{2}$

Step 10

$x = 9$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = 9$ is a local minimum

Step 11

Find the y-value when $x = 9$ .

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Step 11.1

Replace the variable $x$ with $9$ in the expression.

$f (9) = 2 {(9)}^{\frac{5}{2}} - 135 \cdot 9 - 1$

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

Rewrite $9$ as $3^{2}$ .

$f (9) = 2 {(3^{2})}^{\frac{5}{2}} - 135 \cdot 9 - 1$

Step 11.2.1.2

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

$f (9) = 2 \cdot 3^{2 (\frac{5}{2})} - 135 \cdot 9 - 1$

Step 11.2.1.3

Cancel the common factor of $2$ .

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Step 11.2.1.3.1

Cancel the common factor.

$f (9) = 2 \cdot 3^{2 (\frac{5}{2})} - 135 \cdot 9 - 1$

Step 11.2.1.3.2

Rewrite the expression.

$f (9) = 2 \cdot 3^{5} - 135 \cdot 9 - 1$

Step 11.2.1.4

Raise $3$ to the power of $5$ .

$f (9) = 2 \cdot 243 - 135 \cdot 9 - 1$

Step 11.2.1.5

Multiply $2$ by $243$ .

$f (9) = 486 - 135 \cdot 9 - 1$

Step 11.2.1.6

Multiply $- 135$ by $9$ .

$f (9) = 486 - 1215 - 1$

Step 11.2.2

Simplify by subtracting numbers.

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Step 11.2.2.1

Subtract $1215$ from $486$ .

$f (9) = - 729 - 1$

Step 11.2.2.2

Subtract $1$ from $- 729$ .

$f (9) = - 730$

Step 11.2.3

The final answer is $- 730$ .

$y = - 730$

Step 12

These are the local extrema for $f (x) = 2 x^{\frac{5}{2}} - 135 x - 1$ .

$(9, - 730)$ is a local minima

Step 13