Calculus Examples

$f (x) = - 2 x^{\frac{3}{2}} + 9 x + 15$

Step 1

Find the first derivative of the function.

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Step 1.1

By the Sum Rule, the derivative of $- 2 x^{\frac{3}{2}} + 9 x + 15$ with respect to $x$ is $\frac{d}{d x} [- 2 x^{\frac{3}{2}}] + \frac{d}{d x} [9 x] + \frac{d}{d x} [15]$ .

$\frac{d}{d x} [- 2 x^{\frac{3}{2}}] + \frac{d}{d x} [9 x] + \frac{d}{d x} [15]$

Step 1.2

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

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Step 1.2.1

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

$- 2 \frac{d}{d x} [x^{\frac{3}{2}}] + \frac{d}{d x} [9 x] + \frac{d}{d x} [15]$

Step 1.2.2

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

$- 2 (\frac{3}{2} x^{\frac{3}{2} - 1}) + \frac{d}{d x} [9 x] + \frac{d}{d x} [15]$

Step 1.2.3

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

$- 2 (\frac{3}{2} x^{\frac{3}{2} - 1 \cdot \frac{2}{2}}) + \frac{d}{d x} [9 x] + \frac{d}{d x} [15]$

Step 1.2.4

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

$- 2 (\frac{3}{2} x^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}}) + \frac{d}{d x} [9 x] + \frac{d}{d x} [15]$

Step 1.2.5

Combine the numerators over the common denominator.

$- 2 (\frac{3}{2} x^{\frac{3 - 1 \cdot 2}{2}}) + \frac{d}{d x} [9 x] + \frac{d}{d x} [15]$

Step 1.2.6

Simplify the numerator.

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Step 1.2.6.1

Multiply $- 1$ by $2$ .

$- 2 (\frac{3}{2} x^{\frac{3 - 2}{2}}) + \frac{d}{d x} [9 x] + \frac{d}{d x} [15]$

Step 1.2.6.2

Subtract $2$ from $3$ .

$- 2 (\frac{3}{2} x^{\frac{1}{2}}) + \frac{d}{d x} [9 x] + \frac{d}{d x} [15]$

Step 1.2.7

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

$- 2 \frac{3 x^{\frac{1}{2}}}{2} + \frac{d}{d x} [9 x] + \frac{d}{d x} [15]$

Step 1.2.8

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

$\frac{- 2 (3 x^{\frac{1}{2}})}{2} + \frac{d}{d x} [9 x] + \frac{d}{d x} [15]$

Step 1.2.9

Multiply $3$ by $- 2$ .

$\frac{- 6 x^{\frac{1}{2}}}{2} + \frac{d}{d x} [9 x] + \frac{d}{d x} [15]$

Step 1.2.10

Factor $2$ out of $- 6 x^{\frac{1}{2}}$ .

$\frac{2 (- 3 x^{\frac{1}{2}})}{2} + \frac{d}{d x} [9 x] + \frac{d}{d x} [15]$

Step 1.2.11

Cancel the common factors.

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Step 1.2.11.1

Factor $2$ out of $2$ .

$\frac{2 (- 3 x^{\frac{1}{2}})}{2 (1)} + \frac{d}{d x} [9 x] + \frac{d}{d x} [15]$

Step 1.2.11.2

Cancel the common factor.

$\frac{2 (- 3 x^{\frac{1}{2}})}{2 \cdot 1} + \frac{d}{d x} [9 x] + \frac{d}{d x} [15]$

Step 1.2.11.3

Rewrite the expression.

$\frac{- 3 x^{\frac{1}{2}}}{1} + \frac{d}{d x} [9 x] + \frac{d}{d x} [15]$

Step 1.2.11.4

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

$- 3 x^{\frac{1}{2}} + \frac{d}{d x} [9 x] + \frac{d}{d x} [15]$

Step 1.3

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

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Step 1.3.1

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

$- 3 x^{\frac{1}{2}} + 9 \frac{d}{d x} [x] + \frac{d}{d x} [15]$

Step 1.3.2

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

$- 3 x^{\frac{1}{2}} + 9 \cdot 1 + \frac{d}{d x} [15]$

Step 1.3.3

Multiply $9$ by $1$ .

$- 3 x^{\frac{1}{2}} + 9 + \frac{d}{d x} [15]$

Step 1.4

Differentiate using the Constant Rule.

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Step 1.4.1

Since $15$ is constant with respect to $x$ , the derivative of $15$ with respect to $x$ is $0$ .

$- 3 x^{\frac{1}{2}} + 9 + 0$

Step 1.4.2

Add $- 3 x^{\frac{1}{2}} + 9$ and $0$ .

$- 3 x^{\frac{1}{2}} + 9$

Step 2

Find the second derivative of the function.

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Step 2.1

By the Sum Rule, the derivative of $- 3 x^{\frac{1}{2}} + 9$ with respect to $x$ is $\frac{d}{d x} [- 3 x^{\frac{1}{2}}] + \frac{d}{d x} [9]$ .

$f'' (x) = \frac{d}{d x} (- 3 x^{\frac{1}{2}}) + \frac{d}{d x} (9)$

Step 2.2

Evaluate $\frac{d}{d x} [- 3 x^{\frac{1}{2}}]$ .

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Step 2.2.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x^{\frac{1}{2}}$ with respect to $x$ is $- 3 \frac{d}{d x} [x^{\frac{1}{2}}]$ .

$f'' (x) = - 3 \frac{d}{d x} x^{\frac{1}{2}} + \frac{d}{d x} (9)$

Step 2.2.2

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

$f'' (x) = - 3 (\frac{1}{2} x^{\frac{1}{2} - 1}) + \frac{d}{d x} (9)$

Step 2.2.3

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

$f'' (x) = - 3 (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}}) + \frac{d}{d x} (9)$

Step 2.2.4

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

$f'' (x) = - 3 (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}) + \frac{d}{d x} (9)$

Step 2.2.5

Combine the numerators over the common denominator.

$f'' (x) = - 3 (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}}) + \frac{d}{d x} (9)$

Step 2.2.6

Simplify the numerator.

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Step 2.2.6.1

Multiply $- 1$ by $2$ .

$f'' (x) = - 3 (\frac{1}{2} x^{\frac{1 - 2}{2}}) + \frac{d}{d x} (9)$

Step 2.2.6.2

Subtract $2$ from $1$ .

$f'' (x) = - 3 (\frac{1}{2} x^{\frac{- 1}{2}}) + \frac{d}{d x} (9)$

Step 2.2.7

Move the negative in front of the fraction.

$f'' (x) = - 3 (\frac{1}{2} x^{- \frac{1}{2}}) + \frac{d}{d x} (9)$

Step 2.2.8

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

$f'' (x) = - 3 \frac{x^{- \frac{1}{2}}}{2} + \frac{d}{d x} (9)$

Step 2.2.9

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

$f'' (x) = \frac{- 3 x^{- \frac{1}{2}}}{2} + \frac{d}{d x} (9)$

Step 2.2.10

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

$f'' (x) = \frac{- 3}{2 x^{\frac{1}{2}}} + \frac{d}{d x} (9)$

Step 2.2.11

Move the negative in front of the fraction.

$f'' (x) = - \frac{3}{2 x^{\frac{1}{2}}} + \frac{d}{d x} (9)$

Step 2.3

Differentiate using the Constant Rule.

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Step 2.3.1

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$f'' (x) = - \frac{3}{2 x^{\frac{1}{2}}} + 0$

Step 2.3.2

Add $- \frac{3}{2 x^{\frac{1}{2}}}$ and $0$ .

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

Step 3

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

$- 3 x^{\frac{1}{2}} + 9 = 0$

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 $- 2 x^{\frac{3}{2}} + 9 x + 15$ with respect to $x$ is $\frac{d}{d x} [- 2 x^{\frac{3}{2}}] + \frac{d}{d x} [9 x] + \frac{d}{d x} [15]$ .

$\frac{d}{d x} [- 2 x^{\frac{3}{2}}] + \frac{d}{d x} [9 x] + \frac{d}{d x} [15]$

Step 4.1.2

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

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Step 4.1.2.1

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

$- 2 \frac{d}{d x} [x^{\frac{3}{2}}] + \frac{d}{d x} [9 x] + \frac{d}{d x} [15]$

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 $n = \frac{3}{2}$ .

$- 2 (\frac{3}{2} x^{\frac{3}{2} - 1}) + \frac{d}{d x} [9 x] + \frac{d}{d x} [15]$

Step 4.1.2.3

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

$- 2 (\frac{3}{2} x^{\frac{3}{2} - 1 \cdot \frac{2}{2}}) + \frac{d}{d x} [9 x] + \frac{d}{d x} [15]$

Step 4.1.2.4

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

$- 2 (\frac{3}{2} x^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}}) + \frac{d}{d x} [9 x] + \frac{d}{d x} [15]$

Step 4.1.2.5

Combine the numerators over the common denominator.

$- 2 (\frac{3}{2} x^{\frac{3 - 1 \cdot 2}{2}}) + \frac{d}{d x} [9 x] + \frac{d}{d x} [15]$

Step 4.1.2.6

Simplify the numerator.

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Step 4.1.2.6.1

Multiply $- 1$ by $2$ .

$- 2 (\frac{3}{2} x^{\frac{3 - 2}{2}}) + \frac{d}{d x} [9 x] + \frac{d}{d x} [15]$

Step 4.1.2.6.2

Subtract $2$ from $3$ .

$- 2 (\frac{3}{2} x^{\frac{1}{2}}) + \frac{d}{d x} [9 x] + \frac{d}{d x} [15]$

Step 4.1.2.7

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

$- 2 \frac{3 x^{\frac{1}{2}}}{2} + \frac{d}{d x} [9 x] + \frac{d}{d x} [15]$

Step 4.1.2.8

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

$\frac{- 2 (3 x^{\frac{1}{2}})}{2} + \frac{d}{d x} [9 x] + \frac{d}{d x} [15]$

Step 4.1.2.9

Multiply $3$ by $- 2$ .

$\frac{- 6 x^{\frac{1}{2}}}{2} + \frac{d}{d x} [9 x] + \frac{d}{d x} [15]$

Step 4.1.2.10

Factor $2$ out of $- 6 x^{\frac{1}{2}}$ .

$\frac{2 (- 3 x^{\frac{1}{2}})}{2} + \frac{d}{d x} [9 x] + \frac{d}{d x} [15]$

Step 4.1.2.11

Cancel the common factors.

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Step 4.1.2.11.1

Factor $2$ out of $2$ .

$\frac{2 (- 3 x^{\frac{1}{2}})}{2 (1)} + \frac{d}{d x} [9 x] + \frac{d}{d x} [15]$

Step 4.1.2.11.2

Cancel the common factor.

$\frac{2 (- 3 x^{\frac{1}{2}})}{2 \cdot 1} + \frac{d}{d x} [9 x] + \frac{d}{d x} [15]$

Step 4.1.2.11.3

Rewrite the expression.

$\frac{- 3 x^{\frac{1}{2}}}{1} + \frac{d}{d x} [9 x] + \frac{d}{d x} [15]$

Step 4.1.2.11.4

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

$- 3 x^{\frac{1}{2}} + \frac{d}{d x} [9 x] + \frac{d}{d x} [15]$

Step 4.1.3

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

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Step 4.1.3.1

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

$- 3 x^{\frac{1}{2}} + 9 \frac{d}{d x} [x] + \frac{d}{d x} [15]$

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 $n = 1$ .

$- 3 x^{\frac{1}{2}} + 9 \cdot 1 + \frac{d}{d x} [15]$

Step 4.1.3.3

Multiply $9$ by $1$ .

$- 3 x^{\frac{1}{2}} + 9 + \frac{d}{d x} [15]$

Step 4.1.4

Differentiate using the Constant Rule.

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Step 4.1.4.1

Since $15$ is constant with respect to $x$ , the derivative of $15$ with respect to $x$ is $0$ .

$- 3 x^{\frac{1}{2}} + 9 + 0$

Step 4.1.4.2

Add $- 3 x^{\frac{1}{2}} + 9$ and $0$ .

$f' (x) = - 3 x^{\frac{1}{2}} + 9$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $- 3 x^{\frac{1}{2}} + 9$ .

$- 3 x^{\frac{1}{2}} + 9$

Step 5

Set the first derivative equal to $0$ then solve the equation $- 3 x^{\frac{1}{2}} + 9 = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$- 3 x^{\frac{1}{2}} + 9 = 0$

Step 5.2

Subtract $9$ from both sides of the equation.

$- 3 x^{\frac{1}{2}} = - 9$

Step 5.3

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

${(- 3 x^{\frac{1}{2}})}^{2} = {(- 9)}^{2}$

Step 5.4

Simplify the exponent.

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Step 5.4.1

Simplify the left side.

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Step 5.4.1.1

Simplify ${(- 3 x^{\frac{1}{2}})}^{2}$ .

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Step 5.4.1.1.1

Apply the product rule to $- 3 x^{\frac{1}{2}}$ .

${(- 3)}^{2} {(x^{\frac{1}{2}})}^{2} = {(- 9)}^{2}$

Step 5.4.1.1.2

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

$9 {(x^{\frac{1}{2}})}^{2} = {(- 9)}^{2}$

Step 5.4.1.1.3

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

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Step 5.4.1.1.3.1

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

$9 x^{\frac{1}{2} \cdot 2} = {(- 9)}^{2}$

Step 5.4.1.1.3.2

Cancel the common factor of $2$ .

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Step 5.4.1.1.3.2.1

Cancel the common factor.

$9 x^{\frac{1}{2} \cdot 2} = {(- 9)}^{2}$

Step 5.4.1.1.3.2.2

Rewrite the expression.

$9 x^{1} = {(- 9)}^{2}$

Step 5.4.1.1.4

Simplify.

$9 x = {(- 9)}^{2}$

Step 5.4.2

Simplify the right side.

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Step 5.4.2.1

Raise $- 9$ to the power of $2$ .

$9 x = 81$

Step 5.5

Divide each term in $9 x = 81$ by $9$ and simplify.

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Step 5.5.1

Divide each term in $9 x = 81$ by $9$ .

$\frac{9 x}{9} = \frac{81}{9}$

Step 5.5.2

Simplify the left side.

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Step 5.5.2.1

Cancel the common factor of $9$ .

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Step 5.5.2.1.1

Cancel the common factor.

$\frac{9 x}{9} = \frac{81}{9}$

Step 5.5.2.1.2

Divide $x$ by $1$ .

$x = \frac{81}{9}$

Step 5.5.3

Simplify the right side.

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Step 5.5.3.1

Divide $81$ by $9$ .

$x = 9$

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.

$- 3 \sqrt{x^{1}} + 9$

Step 6.1.2

Anything raised to $1$ is the base itself.

$- 3 \sqrt{x} + 9$

Step 6.2

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

$x < 0$

Step 6.3

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.

$- \frac{3}{2 {(9)}^{\frac{1}{2}}}$

Step 9

Evaluate the second derivative.

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Step 9.1

Simplify the denominator.

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Step 9.1.1

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

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

Step 9.1.2

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

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

Step 9.1.3

Cancel the common factor of $2$ .

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Step 9.1.3.1

Cancel the common factor.

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

Step 9.1.3.2

Rewrite the expression.

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

Step 9.1.4

Evaluate the exponent.

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

Step 9.2

Reduce the expression by cancelling the common factors.

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Step 9.2.1

Multiply $2$ by $3$ .

$- \frac{3}{6}$

Step 9.2.2

Cancel the common factor of $3$ and $6$ .

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Step 9.2.2.1

Factor $3$ out of $3$ .

$- \frac{3 (1)}{6}$

Step 9.2.2.2

Cancel the common factors.

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Step 9.2.2.2.1

Factor $3$ out of $6$ .

$- \frac{3 \cdot 1}{3 \cdot 2}$

Step 9.2.2.2.2

Cancel the common factor.

$- \frac{3 \cdot 1}{3 \cdot 2}$

Step 9.2.2.2.3

Rewrite the expression.

$- \frac{1}{2}$

Step 10

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

$x = 9$ is a local maximum

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.

$f (9) = - 2 {(9)}^{\frac{3}{2}} + 9 (9) + 15$

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 $3^{2}$ .

$f (9) = - 2 {(3^{2})}^{\frac{3}{2}} + 9 (9) + 15$

Step 11.2.1.2

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

$f (9) = - 2 \cdot 3^{2 (\frac{3}{2})} + 9 (9) + 15$

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.

$f (9) = - 2 \cdot 3^{2 (\frac{3}{2})} + 9 (9) + 15$

Step 11.2.1.3.2

Rewrite the expression.

$f (9) = - 2 \cdot 3^{3} + 9 (9) + 15$

Step 11.2.1.4

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

$f (9) = - 2 \cdot 27 + 9 (9) + 15$

Step 11.2.1.5

Multiply $- 2$ by $27$ .

$f (9) = - 54 + 9 (9) + 15$

Step 11.2.1.6

Multiply $9$ by $9$ .

$f (9) = - 54 + 81 + 15$

Step 11.2.2

Simplify by adding numbers.

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Step 11.2.2.1

Add $- 54$ and $81$ .

$f (9) = 27 + 15$

Step 11.2.2.2

Add $27$ and $15$ .

$f (9) = 42$

Step 11.2.3

The final answer is $42$ .

$y = 42$

Step 12

These are the local extrema for $f (x) = - 2 x^{\frac{3}{2}} + 9 x + 15$ .

$(9, 42)$ is a local maxima

Step 13