Calculus Examples

$- x^{\frac{3}{2}} + 6 x + 10$

Step 1

Write $- x^{\frac{3}{2}} + 6 x + 10$ as a function.

$f (x) = - x^{\frac{3}{2}} + 6 x + 10$

Step 2

Find the first derivative of the function.

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Step 2.1

By the Sum Rule, the derivative of $- x^{\frac{3}{2}} + 6 x + 10$ with respect to $x$ is $\frac{d}{d x} [- x^{\frac{3}{2}}] + \frac{d}{d x} [6 x] + \frac{d}{d x} [10]$ .

$\frac{d}{d x} [- x^{\frac{3}{2}}] + \frac{d}{d x} [6 x] + \frac{d}{d x} [10]$

Step 2.2

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

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Step 2.2.1

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

$- \frac{d}{d x} [x^{\frac{3}{2}}] + \frac{d}{d x} [6 x] + \frac{d}{d x} [10]$

Step 2.2.2

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

$- (\frac{3}{2} x^{\frac{3}{2} - 1}) + \frac{d}{d x} [6 x] + \frac{d}{d x} [10]$

Step 2.2.3

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

$- (\frac{3}{2} x^{\frac{3}{2} - 1 \cdot \frac{2}{2}}) + \frac{d}{d x} [6 x] + \frac{d}{d x} [10]$

Step 2.2.4

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

$- (\frac{3}{2} x^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}}) + \frac{d}{d x} [6 x] + \frac{d}{d x} [10]$

Step 2.2.5

Combine the numerators over the common denominator.

$- (\frac{3}{2} x^{\frac{3 - 1 \cdot 2}{2}}) + \frac{d}{d x} [6 x] + \frac{d}{d x} [10]$

Step 2.2.6

Simplify the numerator.

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Step 2.2.6.1

Multiply $- 1$ by $2$ .

$- (\frac{3}{2} x^{\frac{3 - 2}{2}}) + \frac{d}{d x} [6 x] + \frac{d}{d x} [10]$

Step 2.2.6.2

Subtract $2$ from $3$ .

$- (\frac{3}{2} x^{\frac{1}{2}}) + \frac{d}{d x} [6 x] + \frac{d}{d x} [10]$

Step 2.2.7

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

$- \frac{3 x^{\frac{1}{2}}}{2} + \frac{d}{d x} [6 x] + \frac{d}{d x} [10]$

Step 2.3

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

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Step 2.3.1

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

$- \frac{3 x^{\frac{1}{2}}}{2} + 6 \frac{d}{d x} [x] + \frac{d}{d x} [10]$

Step 2.3.2

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

$- \frac{3 x^{\frac{1}{2}}}{2} + 6 \cdot 1 + \frac{d}{d x} [10]$

Step 2.3.3

Multiply $6$ by $1$ .

$- \frac{3 x^{\frac{1}{2}}}{2} + 6 + \frac{d}{d x} [10]$

Step 2.4

Since $10$ is constant with respect to $x$ , the derivative of $10$ with respect to $x$ is $0$ .

$- \frac{3 x^{\frac{1}{2}}}{2} + 6 + 0$

Step 2.5

Simplify.

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Step 2.5.1

Add $- \frac{3 x^{\frac{1}{2}}}{2} + 6$ and $0$ .

$- \frac{3 x^{\frac{1}{2}}}{2} + 6$

Step 2.5.2

Reorder terms.

$6 - \frac{3 x^{\frac{1}{2}}}{2}$

Step 3

Find the second derivative of the function.

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Step 3.1

Differentiate.

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Step 3.1.1

By the Sum Rule, the derivative of $6 - \frac{3 x^{\frac{1}{2}}}{2}$ with respect to $x$ is $\frac{d}{d x} [6] + \frac{d}{d x} [- \frac{3 x^{\frac{1}{2}}}{2}]$ .

$f'' (x) = \frac{d}{d x} (6) + \frac{d}{d x} (- \frac{3 x^{\frac{1}{2}}}{2})$

Step 3.1.2

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.2.4

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

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

Step 3.2.5

Combine the numerators over the common denominator.

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

Step 3.2.6

Simplify the numerator.

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Step 3.2.6.1

Multiply $- 1$ by $2$ .

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

Step 3.2.6.2

Subtract $2$ from $1$ .

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

Step 3.2.7

Move the negative in front of the fraction.

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

Step 3.2.8

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

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

Step 3.2.9

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

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

Step 3.2.10

Multiply $2$ by $2$ .

$f'' (x) = 0 - \frac{x^{- \frac{1}{2}} \cdot 3}{4}$

Step 3.2.11

Move $3$ to the left of $x^{- \frac{1}{2}}$ .

$f'' (x) = 0 - \frac{3 \cdot x^{- \frac{1}{2}}}{4}$

Step 3.2.12

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

$f'' (x) = 0 - \frac{3}{4 x^{\frac{1}{2}}}$

Step 3.3

Subtract $\frac{3}{4 x^{\frac{1}{2}}}$ from $0$ .

$f'' (x) = - \frac{3}{4 x^{\frac{1}{2}}}$

Step 4

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

$6 - \frac{3 x^{\frac{1}{2}}}{2} = 0$

Step 5

Find the first derivative.

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Step 5.1

Find the first derivative.

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Step 5.1.1

By the Sum Rule, the derivative of $- x^{\frac{3}{2}} + 6 x + 10$ with respect to $x$ is $\frac{d}{d x} [- x^{\frac{3}{2}}] + \frac{d}{d x} [6 x] + \frac{d}{d x} [10]$ .

$\frac{d}{d x} [- x^{\frac{3}{2}}] + \frac{d}{d x} [6 x] + \frac{d}{d x} [10]$

Step 5.1.2

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

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Step 5.1.2.1

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

$- \frac{d}{d x} [x^{\frac{3}{2}}] + \frac{d}{d x} [6 x] + \frac{d}{d x} [10]$

Step 5.1.2.2

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

$- (\frac{3}{2} x^{\frac{3}{2} - 1}) + \frac{d}{d x} [6 x] + \frac{d}{d x} [10]$

Step 5.1.2.3

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

$- (\frac{3}{2} x^{\frac{3}{2} - 1 \cdot \frac{2}{2}}) + \frac{d}{d x} [6 x] + \frac{d}{d x} [10]$

Step 5.1.2.4

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

$- (\frac{3}{2} x^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}}) + \frac{d}{d x} [6 x] + \frac{d}{d x} [10]$

Step 5.1.2.5

Combine the numerators over the common denominator.

$- (\frac{3}{2} x^{\frac{3 - 1 \cdot 2}{2}}) + \frac{d}{d x} [6 x] + \frac{d}{d x} [10]$

Step 5.1.2.6

Simplify the numerator.

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Step 5.1.2.6.1

Multiply $- 1$ by $2$ .

$- (\frac{3}{2} x^{\frac{3 - 2}{2}}) + \frac{d}{d x} [6 x] + \frac{d}{d x} [10]$

Step 5.1.2.6.2

Subtract $2$ from $3$ .

$- (\frac{3}{2} x^{\frac{1}{2}}) + \frac{d}{d x} [6 x] + \frac{d}{d x} [10]$

Step 5.1.2.7

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

$- \frac{3 x^{\frac{1}{2}}}{2} + \frac{d}{d x} [6 x] + \frac{d}{d x} [10]$

Step 5.1.3

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

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Step 5.1.3.1

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

$- \frac{3 x^{\frac{1}{2}}}{2} + 6 \frac{d}{d x} [x] + \frac{d}{d x} [10]$

Step 5.1.3.2

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

$- \frac{3 x^{\frac{1}{2}}}{2} + 6 \cdot 1 + \frac{d}{d x} [10]$

Step 5.1.3.3

Multiply $6$ by $1$ .

$- \frac{3 x^{\frac{1}{2}}}{2} + 6 + \frac{d}{d x} [10]$

Step 5.1.4

Since $10$ is constant with respect to $x$ , the derivative of $10$ with respect to $x$ is $0$ .

$- \frac{3 x^{\frac{1}{2}}}{2} + 6 + 0$

Step 5.1.5

Simplify.

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Step 5.1.5.1

Add $- \frac{3 x^{\frac{1}{2}}}{2} + 6$ and $0$ .

$- \frac{3 x^{\frac{1}{2}}}{2} + 6$

Step 5.1.5.2

Reorder terms.

$f' (x) = 6 - \frac{3 x^{\frac{1}{2}}}{2}$

Step 5.2

The first derivative of $f (x)$ with respect to $x$ is $6 - \frac{3 x^{\frac{1}{2}}}{2}$ .

$6 - \frac{3 x^{\frac{1}{2}}}{2}$

Step 6

Set the first derivative equal to $0$ then solve the equation $6 - \frac{3 x^{\frac{1}{2}}}{2} = 0$ .

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Step 6.1

Set the first derivative equal to $0$ .

$6 - \frac{3 x^{\frac{1}{2}}}{2} = 0$

Step 6.2

Subtract $6$ from both sides of the equation.

$- \frac{3 x^{\frac{1}{2}}}{2} = - 6$

Step 6.3

Multiply both sides of the equation by $- \frac{2}{3}$ .

$- \frac{2}{3} (- \frac{3 x^{\frac{1}{2}}}{2}) = - \frac{2}{3} \cdot - 6$

Step 6.4

Simplify both sides of the equation.

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Step 6.4.1

Simplify the left side.

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Step 6.4.1.1

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

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Step 6.4.1.1.1

Cancel the common factor of $2$ .

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Step 6.4.1.1.1.1

Move the leading negative in $- \frac{2}{3}$ into the numerator.

$\frac{- 2}{3} (- \frac{3 x^{\frac{1}{2}}}{2}) = - \frac{2}{3} \cdot - 6$

Step 6.4.1.1.1.2

Move the leading negative in $- \frac{3 x^{\frac{1}{2}}}{2}$ into the numerator.

$\frac{- 2}{3} \cdot \frac{- 3 x^{\frac{1}{2}}}{2} = - \frac{2}{3} \cdot - 6$

Step 6.4.1.1.1.3

Factor $2$ out of $- 2$ .

$\frac{2 (- 1)}{3} \cdot \frac{- 3 x^{\frac{1}{2}}}{2} = - \frac{2}{3} \cdot - 6$

Step 6.4.1.1.1.4

Cancel the common factor.

$\frac{2 \cdot - 1}{3} \cdot \frac{- 3 x^{\frac{1}{2}}}{2} = - \frac{2}{3} \cdot - 6$

Step 6.4.1.1.1.5

Rewrite the expression.

$\frac{- 1}{3} (- 3 x^{\frac{1}{2}}) = - \frac{2}{3} \cdot - 6$

Step 6.4.1.1.2

Cancel the common factor of $3$ .

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Step 6.4.1.1.2.1

Factor $3$ out of $- 3 x^{\frac{1}{2}}$ .

$\frac{- 1}{3} (3 (- x^{\frac{1}{2}})) = - \frac{2}{3} \cdot - 6$

Step 6.4.1.1.2.2

Cancel the common factor.

$\frac{- 1}{3} (3 (- x^{\frac{1}{2}})) = - \frac{2}{3} \cdot - 6$

Step 6.4.1.1.2.3

Rewrite the expression.

$- - x^{\frac{1}{2}} = - \frac{2}{3} \cdot - 6$

Step 6.4.1.1.3

Multiply.

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Step 6.4.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 x^{\frac{1}{2}} = - \frac{2}{3} \cdot - 6$

Step 6.4.1.1.3.2

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

$x^{\frac{1}{2}} = - \frac{2}{3} \cdot - 6$

Step 6.4.2

Simplify the right side.

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Step 6.4.2.1

Simplify $- \frac{2}{3} \cdot - 6$ .

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Step 6.4.2.1.1

Cancel the common factor of $3$ .

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Step 6.4.2.1.1.1

Move the leading negative in $- \frac{2}{3}$ into the numerator.

$x^{\frac{1}{2}} = \frac{- 2}{3} \cdot - 6$

Step 6.4.2.1.1.2

Factor $3$ out of $- 6$ .

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

Step 6.4.2.1.1.3

Cancel the common factor.

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

Step 6.4.2.1.1.4

Rewrite the expression.

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

Step 6.4.2.1.2

Multiply $- 2$ by $- 2$ .

$x^{\frac{1}{2}} = 4$

Step 6.5

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

${(x^{\frac{1}{2}})}^{2} = 4^{2}$

Step 6.6

Simplify the exponent.

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Step 6.6.1

Simplify the left side.

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Step 6.6.1.1

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

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Step 6.6.1.1.1

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

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Step 6.6.1.1.1.1

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

$x^{\frac{1}{2} \cdot 2} = 4^{2}$

Step 6.6.1.1.1.2

Cancel the common factor of $2$ .

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Step 6.6.1.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = 4^{2}$

Step 6.6.1.1.1.2.2

Rewrite the expression.

$x^{1} = 4^{2}$

Step 6.6.1.1.2

Simplify.

$x = 4^{2}$

Step 6.6.2

Simplify the right side.

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Step 6.6.2.1

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

$x = 16$

Step 7

Find the values where the derivative is undefined.

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Step 7.1

Convert expressions with fractional exponents to radicals.

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Step 7.1.1

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

$6 - \frac{3 \sqrt{x^{1}}}{2}$

Step 7.1.2

Anything raised to $1$ is the base itself.

$6 - \frac{3 \sqrt{x}}{2}$

Step 7.2

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

$x < 0$

Step 7.3

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 8

Critical points to evaluate.

$x = 16$

Step 9

Evaluate the second derivative at $x = 16$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- \frac{3}{4 {(16)}^{\frac{1}{2}}}$

Step 10

Evaluate the second derivative.

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Step 10.1

Simplify the denominator.

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Step 10.1.1

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

$- \frac{3}{2^{2} \cdot 16^{\frac{1}{2}}}$

Step 10.1.2

Rewrite $16$ as $2^{4}$ .

$- \frac{3}{2^{2} \cdot {(2^{4})}^{\frac{1}{2}}}$

Step 10.1.3

Multiply the exponents in ${(2^{4})}^{\frac{1}{2}}$ .

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Step 10.1.3.1

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

$- \frac{3}{2^{2} \cdot 2^{4 (\frac{1}{2})}}$

Step 10.1.3.2

Cancel the common factor of $2$ .

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Step 10.1.3.2.1

Factor $2$ out of $4$ .

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

Step 10.1.3.2.2

Cancel the common factor.

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

Step 10.1.3.2.3

Rewrite the expression.

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

Step 10.1.4

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

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

Step 10.1.5

Add $2$ and $2$ .

$- \frac{3}{2^{4}}$

Step 10.2

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

$- \frac{3}{16}$

Step 11

$x = 16$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = 16$ is a local maximum

Step 12

Find the y-value when $x = 16$ .

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Step 12.1

Replace the variable $x$ with $16$ in the expression.

$f (16) = - {(16)}^{\frac{3}{2}} + 6 (16) + 10$

Step 12.2

Simplify the result.

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Step 12.2.1

Simplify each term.

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Step 12.2.1.1

Rewrite $16$ as $4^{2}$ .

$f (16) = - {(4^{2})}^{\frac{3}{2}} + 6 (16) + 10$

Step 12.2.1.2

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

$f (16) = - 4^{2 (\frac{3}{2})} + 6 (16) + 10$

Step 12.2.1.3

Cancel the common factor of $2$ .

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Step 12.2.1.3.1

Cancel the common factor.

$f (16) = - 4^{2 (\frac{3}{2})} + 6 (16) + 10$

Step 12.2.1.3.2

Rewrite the expression.

$f (16) = - 4^{3} + 6 (16) + 10$

Step 12.2.1.4

Raise $4$ to the power of $3$ .

$f (16) = - 1 \cdot 64 + 6 (16) + 10$

Step 12.2.1.5

Multiply $- 1$ by $64$ .

$f (16) = - 64 + 6 (16) + 10$

Step 12.2.1.6

Multiply $6$ by $16$ .

$f (16) = - 64 + 96 + 10$

Step 12.2.2

Simplify by adding numbers.

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Step 12.2.2.1

Add $- 64$ and $96$ .

$f (16) = 32 + 10$

Step 12.2.2.2

Add $32$ and $10$ .

$f (16) = 42$

Step 12.2.3

The final answer is $42$ .

$y = 42$

Step 13

These are the local extrema for $f (x) = - x^{\frac{3}{2}} + 6 x + 10$ .

$(16, 42)$ is a local maxima

Step 14