Calculus Examples

$f (x) = x^{5} - 8 x^{3} + 16 x$

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 $x^{5} - 8 x^{3} + 16 x$ with respect to $x$ is $\frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 8 x^{3}] + \frac{d}{d x} [16 x]$ .

$\frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 8 x^{3}] + \frac{d}{d x} [16 x]$

Step 1.1.2

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

$5 x^{4} + \frac{d}{d x} [- 8 x^{3}] + \frac{d}{d x} [16 x]$

Step 1.2

Evaluate $\frac{d}{d x} [- 8 x^{3}]$ .

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Step 1.2.1

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8 x^{3}$ with respect to $x$ is $- 8 \frac{d}{d x} [x^{3}]$ .

$5 x^{4} - 8 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [16 x]$

Step 1.2.2

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

$5 x^{4} - 8 (3 x^{2}) + \frac{d}{d x} [16 x]$

Step 1.2.3

Multiply $3$ by $- 8$ .

$5 x^{4} - 24 x^{2} + \frac{d}{d x} [16 x]$

Step 1.3

Evaluate $\frac{d}{d x} [16 x]$ .

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Step 1.3.1

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

$5 x^{4} - 24 x^{2} + 16 \frac{d}{d x} [x]$

Step 1.3.2

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

$5 x^{4} - 24 x^{2} + 16 \cdot 1$

Step 1.3.3

Multiply $16$ by $1$ .

$5 x^{4} - 24 x^{2} + 16$

Step 2

Find the second derivative of the function.

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Step 2.1

By the Sum Rule, the derivative of $5 x^{4} - 24 x^{2} + 16$ with respect to $x$ is $\frac{d}{d x} [5 x^{4}] + \frac{d}{d x} [- 24 x^{2}] + \frac{d}{d x} [16]$ .

$f'' (x) = \frac{d}{d x} (5 x^{4}) + \frac{d}{d x} (- 24 x^{2}) + \frac{d}{d x} (16)$

Step 2.2

Evaluate $\frac{d}{d x} [5 x^{4}]$ .

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Step 2.2.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{4}$ with respect to $x$ is $5 \frac{d}{d x} [x^{4}]$ .

$f'' (x) = 5 \frac{d}{d x} (x^{4}) + \frac{d}{d x} (- 24 x^{2}) + \frac{d}{d x} (16)$

Step 2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$f'' (x) = 5 (4 x^{3}) + \frac{d}{d x} (- 24 x^{2}) + \frac{d}{d x} (16)$

Step 2.2.3

Multiply $4$ by $5$ .

$f'' (x) = 20 x^{3} + \frac{d}{d x} (- 24 x^{2}) + \frac{d}{d x} (16)$

Step 2.3

Evaluate $\frac{d}{d x} [- 24 x^{2}]$ .

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Step 2.3.1

Since $- 24$ is constant with respect to $x$ , the derivative of $- 24 x^{2}$ with respect to $x$ is $- 24 \frac{d}{d x} [x^{2}]$ .

$f'' (x) = 20 x^{3} - 24 \frac{d}{d x} x^{2} + \frac{d}{d x} (16)$

Step 2.3.2

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

$f'' (x) = 20 x^{3} - 24 (2 x) + \frac{d}{d x} (16)$

Step 2.3.3

Multiply $2$ by $- 24$ .

$f'' (x) = 20 x^{3} - 48 x + \frac{d}{d x} (16)$

Step 2.4

Differentiate using the Constant Rule.

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Step 2.4.1

Since $16$ is constant with respect to $x$ , the derivative of $16$ with respect to $x$ is $0$ .

$f'' (x) = 20 x^{3} - 48 x + 0$

Step 2.4.2

Add $20 x^{3} - 48 x$ and $0$ .

$f'' (x) = 20 x^{3} - 48 x$

Step 3

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

$5 x^{4} - 24 x^{2} + 16 = 0$

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 $x^{5} - 8 x^{3} + 16 x$ with respect to $x$ is $\frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 8 x^{3}] + \frac{d}{d x} [16 x]$ .

$\frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 8 x^{3}] + \frac{d}{d x} [16 x]$

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 $n = 5$ .

$5 x^{4} + \frac{d}{d x} [- 8 x^{3}] + \frac{d}{d x} [16 x]$

Step 4.1.2

Evaluate $\frac{d}{d x} [- 8 x^{3}]$ .

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Step 4.1.2.1

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8 x^{3}$ with respect to $x$ is $- 8 \frac{d}{d x} [x^{3}]$ .

$5 x^{4} - 8 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [16 x]$

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 $n = 3$ .

$5 x^{4} - 8 (3 x^{2}) + \frac{d}{d x} [16 x]$

Step 4.1.2.3

Multiply $3$ by $- 8$ .

$5 x^{4} - 24 x^{2} + \frac{d}{d x} [16 x]$

Step 4.1.3

Evaluate $\frac{d}{d x} [16 x]$ .

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Step 4.1.3.1

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

$5 x^{4} - 24 x^{2} + 16 \frac{d}{d x} [x]$

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 $n = 1$ .

$5 x^{4} - 24 x^{2} + 16 \cdot 1$

Step 4.1.3.3

Multiply $16$ by $1$ .

$f' (x) = 5 x^{4} - 24 x^{2} + 16$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $5 x^{4} - 24 x^{2} + 16$ .

$5 x^{4} - 24 x^{2} + 16$

Step 5

Set the first derivative equal to $0$ then solve the equation $5 x^{4} - 24 x^{2} + 16 = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$5 x^{4} - 24 x^{2} + 16 = 0$

Step 5.2

Substitute $u = x^{2}$ into the equation. This will make the quadratic formula easy to use.

$5 u^{2} - 24 u + 16 = 0$

$u = x^{2}$

Step 5.3

Factor by grouping.

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Step 5.3.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 5 \cdot 16 = 80$ and whose sum is $b = - 24$ .

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Step 5.3.1.1

Factor $- 24$ out of $- 24 u$ .

$5 u^{2} - 24 u + 16 = 0$

Step 5.3.1.2

Rewrite $- 24$ as $- 4$ plus $- 20$

$5 u^{2} + (- 4 - 20) u + 16 = 0$

Step 5.3.1.3

Apply the distributive property.

$5 u^{2} - 4 u - 20 u + 16 = 0$

Step 5.3.2

Factor out the greatest common factor from each group.

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Step 5.3.2.1

Group the first two terms and the last two terms.

$(5 u^{2} - 4 u) - 20 u + 16 = 0$

Step 5.3.2.2

Factor out the greatest common factor (GCF) from each group.

$u (5 u - 4) - 4 (5 u - 4) = 0$

Step 5.3.3

Factor the polynomial by factoring out the greatest common factor, $5 u - 4$ .

$(5 u - 4) (u - 4) = 0$

Step 5.4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$5 u - 4 = 0$

$u - 4 = 0$

Step 5.5

Set $5 u - 4$ equal to $0$ and solve for $u$ .

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Step 5.5.1

Set $5 u - 4$ equal to $0$ .

$5 u - 4 = 0$

Step 5.5.2

Solve $5 u - 4 = 0$ for $u$ .

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Step 5.5.2.1

Add $4$ to both sides of the equation.

$5 u = 4$

Step 5.5.2.2

Divide each term in $5 u = 4$ by $5$ and simplify.

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Step 5.5.2.2.1

Divide each term in $5 u = 4$ by $5$ .

$\frac{5 u}{5} = \frac{4}{5}$

Step 5.5.2.2.2

Simplify the left side.

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Step 5.5.2.2.2.1

Cancel the common factor of $5$ .

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Step 5.5.2.2.2.1.1

Cancel the common factor.

$\frac{5 u}{5} = \frac{4}{5}$

Step 5.5.2.2.2.1.2

Divide $u$ by $1$ .

$u = \frac{4}{5}$

Step 5.6

Set $u - 4$ equal to $0$ and solve for $u$ .

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Step 5.6.1

Set $u - 4$ equal to $0$ .

$u - 4 = 0$

Step 5.6.2

Add $4$ to both sides of the equation.

$u = 4$

Step 5.7

The final solution is all the values that make $(5 u - 4) (u - 4) = 0$ true.

$u = \frac{4}{5}, 4$

Step 5.8

Substitute the real value of $u = x^{2}$ back into the solved equation.

$x^{2} = \frac{4}{5}$

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

Step 5.9

Solve the first equation for $x$ .

$x^{2} = \frac{4}{5}$

Step 5.10

Solve the equation for $x$ .

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Step 5.10.1

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

$x = \pm \sqrt{\frac{4}{5}}$

Step 5.10.2

Simplify $\pm \sqrt{\frac{4}{5}}$ .

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Step 5.10.2.1

Rewrite $\sqrt{\frac{4}{5}}$ as $\frac{\sqrt{4}}{\sqrt{5}}$ .

$x = \pm \frac{\sqrt{4}}{\sqrt{5}}$

Step 5.10.2.2

Simplify the numerator.

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Step 5.10.2.2.1

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

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

Step 5.10.2.2.2

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

$x = \pm \frac{2}{\sqrt{5}}$

Step 5.10.2.3

Multiply $\frac{2}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$x = \pm \frac{2}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Step 5.10.2.4

Combine and simplify the denominator.

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Step 5.10.2.4.1

Multiply $\frac{2}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$x = \pm \frac{2 \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 5.10.2.4.2

Raise $\sqrt{5}$ to the power of $1$ .

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

Step 5.10.2.4.3

Raise $\sqrt{5}$ to the power of $1$ .

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

Step 5.10.2.4.4

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

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

Step 5.10.2.4.5

Add $1$ and $1$ .

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

Step 5.10.2.4.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 5.10.2.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$x = \pm \frac{2 \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Step 5.10.2.4.6.2

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

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

Step 5.10.2.4.6.3

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

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

Step 5.10.2.4.6.4

Cancel the common factor of $2$ .

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Step 5.10.2.4.6.4.1

Cancel the common factor.

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

Step 5.10.2.4.6.4.2

Rewrite the expression.

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

Step 5.10.2.4.6.5

Evaluate the exponent.

$x = \pm \frac{2 \sqrt{5}}{5}$

Step 5.10.3

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

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Step 5.10.3.1

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

$x = \frac{2 \sqrt{5}}{5}$

Step 5.10.3.2

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

$x = - \frac{2 \sqrt{5}}{5}$

Step 5.10.3.3

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

$x = \frac{2 \sqrt{5}}{5}, - \frac{2 \sqrt{5}}{5}$

Step 5.11

Solve the second equation for $x$ .

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

Step 5.12

Solve the equation for $x$ .

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Step 5.12.1

Remove parentheses.

$x^{2} = 4$

Step 5.12.2

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

$x = \pm \sqrt{4}$

Step 5.12.3

Simplify $\pm \sqrt{4}$ .

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Step 5.12.3.1

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

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

Step 5.12.3.2

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

$x = \pm 2$

Step 5.12.4

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

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Step 5.12.4.1

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

$x = 2$

Step 5.12.4.2

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

$x = - 2$

Step 5.12.4.3

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

$x = 2, - 2$

Step 5.13

The solution to $5 x^{4} - 24 x^{2} + 16 = 0$ is $x = \frac{2 \sqrt{5}}{5}, - \frac{2 \sqrt{5}}{5}, 2, - 2$ .

$x = \frac{2 \sqrt{5}}{5}, - \frac{2 \sqrt{5}}{5}, 2, - 2$

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.

$x = \frac{2 \sqrt{5}}{5}, - \frac{2 \sqrt{5}}{5}, 2, - 2$

Step 8

Evaluate the second derivative at $x = \frac{2 \sqrt{5}}{5}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$20 {(\frac{2 \sqrt{5}}{5})}^{3} - 48 \frac{2 \sqrt{5}}{5}$

Step 9

Evaluate the second derivative.

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Step 9.1

Simplify each term.

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Step 9.1.1

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

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Step 9.1.1.1

Apply the product rule to $\frac{2 \sqrt{5}}{5}$ .

$20 \frac{{(2 \sqrt{5})}^{3}}{5^{3}} - 48 \frac{2 \sqrt{5}}{5}$

Step 9.1.1.2

Apply the product rule to $2 \sqrt{5}$ .

$20 \frac{2^{3} {\sqrt{5}}^{3}}{5^{3}} - 48 \frac{2 \sqrt{5}}{5}$

Step 9.1.2

Simplify the numerator.

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Step 9.1.2.1

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

$20 \frac{8 {\sqrt{5}}^{3}}{5^{3}} - 48 \frac{2 \sqrt{5}}{5}$

Step 9.1.2.2

Rewrite ${\sqrt{5}}^{3}$ as $\sqrt{5^{3}}$ .

$20 \frac{8 \sqrt{5^{3}}}{5^{3}} - 48 \frac{2 \sqrt{5}}{5}$

Step 9.1.2.3

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

$20 \frac{8 \sqrt{125}}{5^{3}} - 48 \frac{2 \sqrt{5}}{5}$

Step 9.1.2.4

Rewrite $125$ as $5^{2} \cdot 5$ .

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Step 9.1.2.4.1

Factor $25$ out of $125$ .

$20 \frac{8 \sqrt{25 (5)}}{5^{3}} - 48 \frac{2 \sqrt{5}}{5}$

Step 9.1.2.4.2

Rewrite $25$ as $5^{2}$ .

$20 \frac{8 \sqrt{5^{2} \cdot 5}}{5^{3}} - 48 \frac{2 \sqrt{5}}{5}$

Step 9.1.2.5

Pull terms out from under the radical.

$20 \frac{8 \cdot 5 \sqrt{5}}{5^{3}} - 48 \frac{2 \sqrt{5}}{5}$

Step 9.1.2.6

Multiply $8$ by $5$ .

$20 \frac{40 \sqrt{5}}{5^{3}} - 48 \frac{2 \sqrt{5}}{5}$

Step 9.1.3

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

$20 \frac{40 \sqrt{5}}{125} - 48 \frac{2 \sqrt{5}}{5}$

Step 9.1.4

Cancel the common factor of $5$ .

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Step 9.1.4.1

Factor $5$ out of $20$ .

$5 (4) \frac{40 \sqrt{5}}{125} - 48 \frac{2 \sqrt{5}}{5}$

Step 9.1.4.2

Factor $5$ out of $125$ .

$5 \cdot 4 \frac{40 \sqrt{5}}{5 \cdot 25} - 48 \frac{2 \sqrt{5}}{5}$

Step 9.1.4.3

Cancel the common factor.

$5 \cdot 4 \frac{40 \sqrt{5}}{5 \cdot 25} - 48 \frac{2 \sqrt{5}}{5}$

Step 9.1.4.4

Rewrite the expression.

$4 \frac{40 \sqrt{5}}{25} - 48 \frac{2 \sqrt{5}}{5}$

Step 9.1.5

Combine $4$ and $\frac{40 \sqrt{5}}{25}$ .

$\frac{4 (40 \sqrt{5})}{25} - 48 \frac{2 \sqrt{5}}{5}$

Step 9.1.6

Multiply $40$ by $4$ .

$\frac{160 \sqrt{5}}{25} - 48 \frac{2 \sqrt{5}}{5}$

Step 9.1.7

Cancel the common factor of $160$ and $25$ .

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Step 9.1.7.1

Factor $5$ out of $160 \sqrt{5}$ .

$\frac{5 (32 \sqrt{5})}{25} - 48 \frac{2 \sqrt{5}}{5}$

Step 9.1.7.2

Cancel the common factors.

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Step 9.1.7.2.1

Factor $5$ out of $25$ .

$\frac{5 (32 \sqrt{5})}{5 (5)} - 48 \frac{2 \sqrt{5}}{5}$

Step 9.1.7.2.2

Cancel the common factor.

$\frac{5 (32 \sqrt{5})}{5 \cdot 5} - 48 \frac{2 \sqrt{5}}{5}$

Step 9.1.7.2.3

Rewrite the expression.

$\frac{32 \sqrt{5}}{5} - 48 \frac{2 \sqrt{5}}{5}$

Step 9.1.8

Multiply $- 48 \frac{2 \sqrt{5}}{5}$ .

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Step 9.1.8.1

Combine $- 48$ and $\frac{2 \sqrt{5}}{5}$ .

$\frac{32 \sqrt{5}}{5} + \frac{- 48 (2 \sqrt{5})}{5}$

Step 9.1.8.2

Multiply $2$ by $- 48$ .

$\frac{32 \sqrt{5}}{5} + \frac{- 96 \sqrt{5}}{5}$

Step 9.1.9

Move the negative in front of the fraction.

$\frac{32 \sqrt{5}}{5} - \frac{96 \sqrt{5}}{5}$

Step 9.2

Simplify terms.

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Step 9.2.1

Combine the numerators over the common denominator.

$\frac{32 \sqrt{5} - 96 \sqrt{5}}{5}$

Step 9.2.2

Subtract $96 \sqrt{5}$ from $32 \sqrt{5}$ .

$\frac{- 64 \sqrt{5}}{5}$

Step 9.2.3

Move the negative in front of the fraction.

$- \frac{64 \sqrt{5}}{5}$

Step 10

$x = \frac{2 \sqrt{5}}{5}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = \frac{2 \sqrt{5}}{5}$ is a local maximum

Step 11

Find the y-value when $x = \frac{2 \sqrt{5}}{5}$ .

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Step 11.1

Replace the variable $x$ with $\frac{2 \sqrt{5}}{5}$ in the expression.

$f (\frac{2 \sqrt{5}}{5}) = {(\frac{2 \sqrt{5}}{5})}^{5} - 8 {(\frac{2 \sqrt{5}}{5})}^{3} + 16 (\frac{2 \sqrt{5}}{5})$

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

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

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Step 11.2.1.1.1

Apply the product rule to $\frac{2 \sqrt{5}}{5}$ .

$f (\frac{2 \sqrt{5}}{5}) = \frac{{(2 \sqrt{5})}^{5}}{5^{5}} - 8 {(\frac{2 \sqrt{5}}{5})}^{3} + 16 (\frac{2 \sqrt{5}}{5})$

Step 11.2.1.1.2

Apply the product rule to $2 \sqrt{5}$ .

$f (\frac{2 \sqrt{5}}{5}) = \frac{2^{5} {\sqrt{5}}^{5}}{5^{5}} - 8 {(\frac{2 \sqrt{5}}{5})}^{3} + 16 (\frac{2 \sqrt{5}}{5})$

Step 11.2.1.2

Simplify the numerator.

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Step 11.2.1.2.1

Raise $2$ to the power of $5$ .

$f (\frac{2 \sqrt{5}}{5}) = \frac{32 {\sqrt{5}}^{5}}{5^{5}} - 8 {(\frac{2 \sqrt{5}}{5})}^{3} + 16 (\frac{2 \sqrt{5}}{5})$

Step 11.2.1.2.2

Rewrite ${\sqrt{5}}^{5}$ as $\sqrt{5^{5}}$ .

$f (\frac{2 \sqrt{5}}{5}) = \frac{32 \sqrt{5^{5}}}{5^{5}} - 8 {(\frac{2 \sqrt{5}}{5})}^{3} + 16 (\frac{2 \sqrt{5}}{5})$

Step 11.2.1.2.3

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

$f (\frac{2 \sqrt{5}}{5}) = \frac{32 \sqrt{3125}}{5^{5}} - 8 {(\frac{2 \sqrt{5}}{5})}^{3} + 16 (\frac{2 \sqrt{5}}{5})$

Step 11.2.1.2.4

Rewrite $3125$ as $25^{2} \cdot 5$ .

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Step 11.2.1.2.4.1

Factor $625$ out of $3125$ .

$f (\frac{2 \sqrt{5}}{5}) = \frac{32 \sqrt{625 (5)}}{5^{5}} - 8 {(\frac{2 \sqrt{5}}{5})}^{3} + 16 (\frac{2 \sqrt{5}}{5})$

Step 11.2.1.2.4.2

Rewrite $625$ as $25^{2}$ .

$f (\frac{2 \sqrt{5}}{5}) = \frac{32 \sqrt{25^{2} \cdot 5}}{5^{5}} - 8 {(\frac{2 \sqrt{5}}{5})}^{3} + 16 (\frac{2 \sqrt{5}}{5})$

Step 11.2.1.2.5

Pull terms out from under the radical.

$f (\frac{2 \sqrt{5}}{5}) = \frac{32 \cdot (25 \sqrt{5})}{5^{5}} - 8 {(\frac{2 \sqrt{5}}{5})}^{3} + 16 (\frac{2 \sqrt{5}}{5})$

Step 11.2.1.2.6

Multiply $32$ by $25$ .

$f (\frac{2 \sqrt{5}}{5}) = \frac{800 \sqrt{5}}{5^{5}} - 8 {(\frac{2 \sqrt{5}}{5})}^{3} + 16 (\frac{2 \sqrt{5}}{5})$

Step 11.2.1.3

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

$f (\frac{2 \sqrt{5}}{5}) = \frac{800 \sqrt{5}}{3125} - 8 {(\frac{2 \sqrt{5}}{5})}^{3} + 16 (\frac{2 \sqrt{5}}{5})$

Step 11.2.1.4

Cancel the common factor of $800$ and $3125$ .

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Step 11.2.1.4.1

Factor $25$ out of $800 \sqrt{5}$ .

$f (\frac{2 \sqrt{5}}{5}) = \frac{25 (32 \sqrt{5})}{3125} - 8 {(\frac{2 \sqrt{5}}{5})}^{3} + 16 (\frac{2 \sqrt{5}}{5})$

Step 11.2.1.4.2

Cancel the common factors.

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Step 11.2.1.4.2.1

Factor $25$ out of $3125$ .

$f (\frac{2 \sqrt{5}}{5}) = \frac{25 (32 \sqrt{5})}{25 (125)} - 8 {(\frac{2 \sqrt{5}}{5})}^{3} + 16 (\frac{2 \sqrt{5}}{5})$

Step 11.2.1.4.2.2

Cancel the common factor.

$f (\frac{2 \sqrt{5}}{5}) = \frac{25 (32 \sqrt{5})}{25 \cdot 125} - 8 {(\frac{2 \sqrt{5}}{5})}^{3} + 16 (\frac{2 \sqrt{5}}{5})$

Step 11.2.1.4.2.3

Rewrite the expression.

$f (\frac{2 \sqrt{5}}{5}) = \frac{32 \sqrt{5}}{125} - 8 {(\frac{2 \sqrt{5}}{5})}^{3} + 16 (\frac{2 \sqrt{5}}{5})$

Step 11.2.1.5

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

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Step 11.2.1.5.1

Apply the product rule to $\frac{2 \sqrt{5}}{5}$ .

$f (\frac{2 \sqrt{5}}{5}) = \frac{32 \sqrt{5}}{125} - 8 \frac{{(2 \sqrt{5})}^{3}}{5^{3}} + 16 (\frac{2 \sqrt{5}}{5})$

Step 11.2.1.5.2

Apply the product rule to $2 \sqrt{5}$ .

$f (\frac{2 \sqrt{5}}{5}) = \frac{32 \sqrt{5}}{125} - 8 \frac{2^{3} {\sqrt{5}}^{3}}{5^{3}} + 16 (\frac{2 \sqrt{5}}{5})$

Step 11.2.1.6

Simplify the numerator.

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Step 11.2.1.6.1

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

$f (\frac{2 \sqrt{5}}{5}) = \frac{32 \sqrt{5}}{125} - 8 \frac{8 {\sqrt{5}}^{3}}{5^{3}} + 16 (\frac{2 \sqrt{5}}{5})$

Step 11.2.1.6.2

Rewrite ${\sqrt{5}}^{3}$ as $\sqrt{5^{3}}$ .

$f (\frac{2 \sqrt{5}}{5}) = \frac{32 \sqrt{5}}{125} - 8 \frac{8 \sqrt{5^{3}}}{5^{3}} + 16 (\frac{2 \sqrt{5}}{5})$

Step 11.2.1.6.3

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

$f (\frac{2 \sqrt{5}}{5}) = \frac{32 \sqrt{5}}{125} - 8 \frac{8 \sqrt{125}}{5^{3}} + 16 (\frac{2 \sqrt{5}}{5})$

Step 11.2.1.6.4

Rewrite $125$ as $5^{2} \cdot 5$ .

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Step 11.2.1.6.4.1

Factor $25$ out of $125$ .

$f (\frac{2 \sqrt{5}}{5}) = \frac{32 \sqrt{5}}{125} - 8 \frac{8 \sqrt{25 (5)}}{5^{3}} + 16 (\frac{2 \sqrt{5}}{5})$

Step 11.2.1.6.4.2

Rewrite $25$ as $5^{2}$ .

$f (\frac{2 \sqrt{5}}{5}) = \frac{32 \sqrt{5}}{125} - 8 \frac{8 \sqrt{5^{2} \cdot 5}}{5^{3}} + 16 (\frac{2 \sqrt{5}}{5})$

Step 11.2.1.6.5

Pull terms out from under the radical.

$f (\frac{2 \sqrt{5}}{5}) = \frac{32 \sqrt{5}}{125} - 8 \frac{8 \cdot (5 \sqrt{5})}{5^{3}} + 16 (\frac{2 \sqrt{5}}{5})$

Step 11.2.1.6.6

Multiply $8$ by $5$ .

$f (\frac{2 \sqrt{5}}{5}) = \frac{32 \sqrt{5}}{125} - 8 \frac{40 \sqrt{5}}{5^{3}} + 16 (\frac{2 \sqrt{5}}{5})$

Step 11.2.1.7

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

$f (\frac{2 \sqrt{5}}{5}) = \frac{32 \sqrt{5}}{125} - 8 \frac{40 \sqrt{5}}{125} + 16 (\frac{2 \sqrt{5}}{5})$

Step 11.2.1.8

Cancel the common factor of $40$ and $125$ .

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Step 11.2.1.8.1

Factor $5$ out of $40 \sqrt{5}$ .

$f (\frac{2 \sqrt{5}}{5}) = \frac{32 \sqrt{5}}{125} - 8 \frac{5 (8 \sqrt{5})}{125} + 16 (\frac{2 \sqrt{5}}{5})$

Step 11.2.1.8.2

Cancel the common factors.

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Step 11.2.1.8.2.1

Factor $5$ out of $125$ .

$f (\frac{2 \sqrt{5}}{5}) = \frac{32 \sqrt{5}}{125} - 8 \frac{5 (8 \sqrt{5})}{5 (25)} + 16 (\frac{2 \sqrt{5}}{5})$

Step 11.2.1.8.2.2

Cancel the common factor.

$f (\frac{2 \sqrt{5}}{5}) = \frac{32 \sqrt{5}}{125} - 8 \frac{5 (8 \sqrt{5})}{5 \cdot 25} + 16 (\frac{2 \sqrt{5}}{5})$

Step 11.2.1.8.2.3

Rewrite the expression.

$f (\frac{2 \sqrt{5}}{5}) = \frac{32 \sqrt{5}}{125} - 8 \frac{8 \sqrt{5}}{25} + 16 (\frac{2 \sqrt{5}}{5})$

Step 11.2.1.9

Multiply $- 8 \frac{8 \sqrt{5}}{25}$ .

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Step 11.2.1.9.1

Combine $- 8$ and $\frac{8 \sqrt{5}}{25}$ .

$f (\frac{2 \sqrt{5}}{5}) = \frac{32 \sqrt{5}}{125} + \frac{- 8 (8 \sqrt{5})}{25} + 16 (\frac{2 \sqrt{5}}{5})$

Step 11.2.1.9.2

Multiply $8$ by $- 8$ .

$f (\frac{2 \sqrt{5}}{5}) = \frac{32 \sqrt{5}}{125} + \frac{- 64 \sqrt{5}}{25} + 16 (\frac{2 \sqrt{5}}{5})$

Step 11.2.1.10

Move the negative in front of the fraction.

$f (\frac{2 \sqrt{5}}{5}) = \frac{32 \sqrt{5}}{125} - \frac{64 \sqrt{5}}{25} + 16 (\frac{2 \sqrt{5}}{5})$

Step 11.2.1.11

Multiply $16 (\frac{2 \sqrt{5}}{5})$ .

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Step 11.2.1.11.1

Combine $16$ and $\frac{2 \sqrt{5}}{5}$ .

$f (\frac{2 \sqrt{5}}{5}) = \frac{32 \sqrt{5}}{125} - \frac{64 \sqrt{5}}{25} + \frac{16 (2 \sqrt{5})}{5}$

Step 11.2.1.11.2

Multiply $2$ by $16$ .

$f (\frac{2 \sqrt{5}}{5}) = \frac{32 \sqrt{5}}{125} - \frac{64 \sqrt{5}}{25} + \frac{32 \sqrt{5}}{5}$

Step 11.2.2

Find the common denominator.

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Step 11.2.2.1

Multiply $\frac{64 \sqrt{5}}{25}$ by $\frac{5}{5}$ .

$f (\frac{2 \sqrt{5}}{5}) = \frac{32 \sqrt{5}}{125} - (\frac{64 \sqrt{5}}{25} \cdot \frac{5}{5}) + \frac{32 \sqrt{5}}{5}$

Step 11.2.2.2

Multiply $\frac{64 \sqrt{5}}{25}$ by $\frac{5}{5}$ .

$f (\frac{2 \sqrt{5}}{5}) = \frac{32 \sqrt{5}}{125} - \frac{64 \sqrt{5} \cdot 5}{25 \cdot 5} + \frac{32 \sqrt{5}}{5}$

Step 11.2.2.3

Multiply $\frac{32 \sqrt{5}}{5}$ by $\frac{25}{25}$ .

$f (\frac{2 \sqrt{5}}{5}) = \frac{32 \sqrt{5}}{125} - \frac{64 \sqrt{5} \cdot 5}{25 \cdot 5} + \frac{32 \sqrt{5}}{5} \cdot \frac{25}{25}$

Step 11.2.2.4

Multiply $\frac{32 \sqrt{5}}{5}$ by $\frac{25}{25}$ .

$f (\frac{2 \sqrt{5}}{5}) = \frac{32 \sqrt{5}}{125} - \frac{64 \sqrt{5} \cdot 5}{25 \cdot 5} + \frac{32 \sqrt{5} \cdot 25}{5 \cdot 25}$

Step 11.2.2.5

Reorder the factors of $25 \cdot 5$ .

$f (\frac{2 \sqrt{5}}{5}) = \frac{32 \sqrt{5}}{125} - \frac{64 \sqrt{5} \cdot 5}{5 \cdot 25} + \frac{32 \sqrt{5} \cdot 25}{5 \cdot 25}$

Step 11.2.2.6

Multiply $5$ by $25$ .

$f (\frac{2 \sqrt{5}}{5}) = \frac{32 \sqrt{5}}{125} - \frac{64 \sqrt{5} \cdot 5}{125} + \frac{32 \sqrt{5} \cdot 25}{5 \cdot 25}$

Step 11.2.2.7

Multiply $5$ by $25$ .

$f (\frac{2 \sqrt{5}}{5}) = \frac{32 \sqrt{5}}{125} - \frac{64 \sqrt{5} \cdot 5}{125} + \frac{32 \sqrt{5} \cdot 25}{125}$

Step 11.2.3

Combine the numerators over the common denominator.

$f (\frac{2 \sqrt{5}}{5}) = \frac{32 \sqrt{5} - 64 \sqrt{5} \cdot 5 + 32 \sqrt{5} \cdot 25}{125}$

Step 11.2.4

Simplify each term.

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Step 11.2.4.1

Multiply $5$ by $- 64$ .

$f (\frac{2 \sqrt{5}}{5}) = \frac{32 \sqrt{5} - 320 \sqrt{5} + 32 \sqrt{5} \cdot 25}{125}$

Step 11.2.4.2

Multiply $25$ by $32$ .

$f (\frac{2 \sqrt{5}}{5}) = \frac{32 \sqrt{5} - 320 \sqrt{5} + 800 \sqrt{5}}{125}$

Step 11.2.5

Simplify by adding terms.

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Step 11.2.5.1

Subtract $320 \sqrt{5}$ from $32 \sqrt{5}$ .

$f (\frac{2 \sqrt{5}}{5}) = \frac{- 288 \sqrt{5} + 800 \sqrt{5}}{125}$

Step 11.2.5.2

Add $- 288 \sqrt{5}$ and $800 \sqrt{5}$ .

$f (\frac{2 \sqrt{5}}{5}) = \frac{512 \sqrt{5}}{125}$

Step 11.2.6

The final answer is $\frac{512 \sqrt{5}}{125}$ .

$y = \frac{512 \sqrt{5}}{125}$

Step 12

Evaluate the second derivative at $x = - \frac{2 \sqrt{5}}{5}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$20 {(- \frac{2 \sqrt{5}}{5})}^{3} - 48 (- \frac{2 \sqrt{5}}{5})$

Step 13

Evaluate the second derivative.

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Step 13.1

Simplify each term.

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Step 13.1.1

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

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Step 13.1.1.1

Apply the product rule to $- \frac{2 \sqrt{5}}{5}$ .

$20 ({(- 1)}^{3} {(\frac{2 \sqrt{5}}{5})}^{3}) - 48 (- \frac{2 \sqrt{5}}{5})$

Step 13.1.1.2

Apply the product rule to $\frac{2 \sqrt{5}}{5}$ .

$20 ({(- 1)}^{3} \frac{{(2 \sqrt{5})}^{3}}{5^{3}}) - 48 (- \frac{2 \sqrt{5}}{5})$

Step 13.1.1.3

Apply the product rule to $2 \sqrt{5}$ .

$20 ({(- 1)}^{3} \frac{2^{3} {\sqrt{5}}^{3}}{5^{3}}) - 48 (- \frac{2 \sqrt{5}}{5})$

Step 13.1.2

Raise $- 1$ to the power of $3$ .

$20 (- \frac{2^{3} {\sqrt{5}}^{3}}{5^{3}}) - 48 (- \frac{2 \sqrt{5}}{5})$

Step 13.1.3

Simplify the numerator.

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Step 13.1.3.1

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

$20 (- \frac{8 {\sqrt{5}}^{3}}{5^{3}}) - 48 (- \frac{2 \sqrt{5}}{5})$

Step 13.1.3.2

Rewrite ${\sqrt{5}}^{3}$ as $\sqrt{5^{3}}$ .

$20 (- \frac{8 \sqrt{5^{3}}}{5^{3}}) - 48 (- \frac{2 \sqrt{5}}{5})$

Step 13.1.3.3

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

$20 (- \frac{8 \sqrt{125}}{5^{3}}) - 48 (- \frac{2 \sqrt{5}}{5})$

Step 13.1.3.4

Rewrite $125$ as $5^{2} \cdot 5$ .

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Step 13.1.3.4.1

Factor $25$ out of $125$ .

$20 (- \frac{8 \sqrt{25 (5)}}{5^{3}}) - 48 (- \frac{2 \sqrt{5}}{5})$

Step 13.1.3.4.2

Rewrite $25$ as $5^{2}$ .

$20 (- \frac{8 \sqrt{5^{2} \cdot 5}}{5^{3}}) - 48 (- \frac{2 \sqrt{5}}{5})$

Step 13.1.3.5

Pull terms out from under the radical.

$20 (- \frac{8 \cdot 5 \sqrt{5}}{5^{3}}) - 48 (- \frac{2 \sqrt{5}}{5})$

Step 13.1.3.6

Multiply $8$ by $5$ .

$20 (- \frac{40 \sqrt{5}}{5^{3}}) - 48 (- \frac{2 \sqrt{5}}{5})$

Step 13.1.4

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

$20 (- \frac{40 \sqrt{5}}{125}) - 48 (- \frac{2 \sqrt{5}}{5})$

Step 13.1.5

Cancel the common factor of $5$ .

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Step 13.1.5.1

Move the leading negative in $- \frac{40 \sqrt{5}}{125}$ into the numerator.

$20 \frac{- 40 \sqrt{5}}{125} - 48 (- \frac{2 \sqrt{5}}{5})$

Step 13.1.5.2

Factor $5$ out of $20$ .

$5 (4) \frac{- 40 \sqrt{5}}{125} - 48 (- \frac{2 \sqrt{5}}{5})$

Step 13.1.5.3

Factor $5$ out of $125$ .

$5 \cdot 4 \frac{- 40 \sqrt{5}}{5 \cdot 25} - 48 (- \frac{2 \sqrt{5}}{5})$

Step 13.1.5.4

Cancel the common factor.

$5 \cdot 4 \frac{- 40 \sqrt{5}}{5 \cdot 25} - 48 (- \frac{2 \sqrt{5}}{5})$

Step 13.1.5.5

Rewrite the expression.

$4 \frac{- 40 \sqrt{5}}{25} - 48 (- \frac{2 \sqrt{5}}{5})$

Step 13.1.6

Combine $4$ and $\frac{- 40 \sqrt{5}}{25}$ .

$\frac{4 (- 40 \sqrt{5})}{25} - 48 (- \frac{2 \sqrt{5}}{5})$

Step 13.1.7

Multiply $- 40$ by $4$ .

$\frac{- 160 \sqrt{5}}{25} - 48 (- \frac{2 \sqrt{5}}{5})$

Step 13.1.8

Cancel the common factor of $- 160$ and $25$ .

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Step 13.1.8.1

Factor $5$ out of $- 160 \sqrt{5}$ .

$\frac{5 (- 32 \sqrt{5})}{25} - 48 (- \frac{2 \sqrt{5}}{5})$

Step 13.1.8.2

Cancel the common factors.

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Step 13.1.8.2.1

Factor $5$ out of $25$ .

$\frac{5 (- 32 \sqrt{5})}{5 (5)} - 48 (- \frac{2 \sqrt{5}}{5})$

Step 13.1.8.2.2

Cancel the common factor.

$\frac{5 (- 32 \sqrt{5})}{5 \cdot 5} - 48 (- \frac{2 \sqrt{5}}{5})$

Step 13.1.8.2.3

Rewrite the expression.

$\frac{- 32 \sqrt{5}}{5} - 48 (- \frac{2 \sqrt{5}}{5})$

Step 13.1.9

Move the negative in front of the fraction.

$- \frac{32 \sqrt{5}}{5} - 48 (- \frac{2 \sqrt{5}}{5})$

Step 13.1.10

Multiply $- 48 (- \frac{2 \sqrt{5}}{5})$ .

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Step 13.1.10.1

Multiply $- 1$ by $- 48$ .

$- \frac{32 \sqrt{5}}{5} + 48 \frac{2 \sqrt{5}}{5}$

Step 13.1.10.2

Combine $48$ and $\frac{2 \sqrt{5}}{5}$ .

$- \frac{32 \sqrt{5}}{5} + \frac{48 (2 \sqrt{5})}{5}$

Step 13.1.10.3

Multiply $2$ by $48$ .

$- \frac{32 \sqrt{5}}{5} + \frac{96 \sqrt{5}}{5}$

Step 13.2

Simplify terms.

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Step 13.2.1

Combine the numerators over the common denominator.

$\frac{- 32 \sqrt{5} + 96 \sqrt{5}}{5}$

Step 13.2.2

Add $- 32 \sqrt{5}$ and $96 \sqrt{5}$ .

$\frac{64 \sqrt{5}}{5}$

Step 14

$x = - \frac{2 \sqrt{5}}{5}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = - \frac{2 \sqrt{5}}{5}$ is a local minimum

Step 15

Find the y-value when $x = - \frac{2 \sqrt{5}}{5}$ .

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Step 15.1

Replace the variable $x$ with $- \frac{2 \sqrt{5}}{5}$ in the expression.

$f (- \frac{2 \sqrt{5}}{5}) = {(- \frac{2 \sqrt{5}}{5})}^{5} - 8 {(- \frac{2 \sqrt{5}}{5})}^{3} + 16 (- \frac{2 \sqrt{5}}{5})$

Step 15.2

Simplify the result.

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Step 15.2.1

Simplify each term.

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Step 15.2.1.1

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

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Step 15.2.1.1.1

Apply the product rule to $- \frac{2 \sqrt{5}}{5}$ .

$f (- \frac{2 \sqrt{5}}{5}) = {(- 1)}^{5} {(\frac{2 \sqrt{5}}{5})}^{5} - 8 {(- \frac{2 \sqrt{5}}{5})}^{3} + 16 (- \frac{2 \sqrt{5}}{5})$

Step 15.2.1.1.2

Apply the product rule to $\frac{2 \sqrt{5}}{5}$ .

$f (- \frac{2 \sqrt{5}}{5}) = {(- 1)}^{5} (\frac{{(2 \sqrt{5})}^{5}}{5^{5}}) - 8 {(- \frac{2 \sqrt{5}}{5})}^{3} + 16 (- \frac{2 \sqrt{5}}{5})$

Step 15.2.1.1.3

Apply the product rule to $2 \sqrt{5}$ .

$f (- \frac{2 \sqrt{5}}{5}) = {(- 1)}^{5} (\frac{2^{5} {\sqrt{5}}^{5}}{5^{5}}) - 8 {(- \frac{2 \sqrt{5}}{5})}^{3} + 16 (- \frac{2 \sqrt{5}}{5})$

Step 15.2.1.2

Raise $- 1$ to the power of $5$ .

$f (- \frac{2 \sqrt{5}}{5}) = - \frac{2^{5} {\sqrt{5}}^{5}}{5^{5}} - 8 {(- \frac{2 \sqrt{5}}{5})}^{3} + 16 (- \frac{2 \sqrt{5}}{5})$

Step 15.2.1.3

Simplify the numerator.

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Step 15.2.1.3.1

Raise $2$ to the power of $5$ .

$f (- \frac{2 \sqrt{5}}{5}) = - \frac{32 {\sqrt{5}}^{5}}{5^{5}} - 8 {(- \frac{2 \sqrt{5}}{5})}^{3} + 16 (- \frac{2 \sqrt{5}}{5})$

Step 15.2.1.3.2

Rewrite ${\sqrt{5}}^{5}$ as $\sqrt{5^{5}}$ .

$f (- \frac{2 \sqrt{5}}{5}) = - \frac{32 \sqrt{5^{5}}}{5^{5}} - 8 {(- \frac{2 \sqrt{5}}{5})}^{3} + 16 (- \frac{2 \sqrt{5}}{5})$

Step 15.2.1.3.3

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

$f (- \frac{2 \sqrt{5}}{5}) = - \frac{32 \sqrt{3125}}{5^{5}} - 8 {(- \frac{2 \sqrt{5}}{5})}^{3} + 16 (- \frac{2 \sqrt{5}}{5})$

Step 15.2.1.3.4

Rewrite $3125$ as $25^{2} \cdot 5$ .

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Step 15.2.1.3.4.1

Factor $625$ out of $3125$ .

$f (- \frac{2 \sqrt{5}}{5}) = - \frac{32 \sqrt{625 (5)}}{5^{5}} - 8 {(- \frac{2 \sqrt{5}}{5})}^{3} + 16 (- \frac{2 \sqrt{5}}{5})$

Step 15.2.1.3.4.2

Rewrite $625$ as $25^{2}$ .

$f (- \frac{2 \sqrt{5}}{5}) = - \frac{32 \sqrt{25^{2} \cdot 5}}{5^{5}} - 8 {(- \frac{2 \sqrt{5}}{5})}^{3} + 16 (- \frac{2 \sqrt{5}}{5})$

Step 15.2.1.3.5

Pull terms out from under the radical.

$f (- \frac{2 \sqrt{5}}{5}) = - \frac{32 \cdot (25 \sqrt{5})}{5^{5}} - 8 {(- \frac{2 \sqrt{5}}{5})}^{3} + 16 (- \frac{2 \sqrt{5}}{5})$

Step 15.2.1.3.6

Multiply $32$ by $25$ .

$f (- \frac{2 \sqrt{5}}{5}) = - \frac{800 \sqrt{5}}{5^{5}} - 8 {(- \frac{2 \sqrt{5}}{5})}^{3} + 16 (- \frac{2 \sqrt{5}}{5})$

Step 15.2.1.4

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

$f (- \frac{2 \sqrt{5}}{5}) = - \frac{800 \sqrt{5}}{3125} - 8 {(- \frac{2 \sqrt{5}}{5})}^{3} + 16 (- \frac{2 \sqrt{5}}{5})$

Step 15.2.1.5

Cancel the common factor of $800$ and $3125$ .

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Step 15.2.1.5.1

Factor $25$ out of $800 \sqrt{5}$ .

$f (- \frac{2 \sqrt{5}}{5}) = - \frac{25 (32 \sqrt{5})}{3125} - 8 {(- \frac{2 \sqrt{5}}{5})}^{3} + 16 (- \frac{2 \sqrt{5}}{5})$

Step 15.2.1.5.2

Cancel the common factors.

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Step 15.2.1.5.2.1

Factor $25$ out of $3125$ .

$f (- \frac{2 \sqrt{5}}{5}) = - \frac{25 (32 \sqrt{5})}{25 (125)} - 8 {(- \frac{2 \sqrt{5}}{5})}^{3} + 16 (- \frac{2 \sqrt{5}}{5})$

Step 15.2.1.5.2.2

Cancel the common factor.

$f (- \frac{2 \sqrt{5}}{5}) = - \frac{25 (32 \sqrt{5})}{25 \cdot 125} - 8 {(- \frac{2 \sqrt{5}}{5})}^{3} + 16 (- \frac{2 \sqrt{5}}{5})$

Step 15.2.1.5.2.3

Rewrite the expression.

$f (- \frac{2 \sqrt{5}}{5}) = - \frac{32 \sqrt{5}}{125} - 8 {(- \frac{2 \sqrt{5}}{5})}^{3} + 16 (- \frac{2 \sqrt{5}}{5})$

Step 15.2.1.6

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

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Step 15.2.1.6.1

Apply the product rule to $- \frac{2 \sqrt{5}}{5}$ .

$f (- \frac{2 \sqrt{5}}{5}) = - \frac{32 \sqrt{5}}{125} - 8 ({(- 1)}^{3} {(\frac{2 \sqrt{5}}{5})}^{3}) + 16 (- \frac{2 \sqrt{5}}{5})$

Step 15.2.1.6.2

Apply the product rule to $\frac{2 \sqrt{5}}{5}$ .

$f (- \frac{2 \sqrt{5}}{5}) = - \frac{32 \sqrt{5}}{125} - 8 ({(- 1)}^{3} (\frac{{(2 \sqrt{5})}^{3}}{5^{3}})) + 16 (- \frac{2 \sqrt{5}}{5})$

Step 15.2.1.6.3

Apply the product rule to $2 \sqrt{5}$ .

$f (- \frac{2 \sqrt{5}}{5}) = - \frac{32 \sqrt{5}}{125} - 8 ({(- 1)}^{3} (\frac{2^{3} {\sqrt{5}}^{3}}{5^{3}})) + 16 (- \frac{2 \sqrt{5}}{5})$

Step 15.2.1.7

Raise $- 1$ to the power of $3$ .

$f (- \frac{2 \sqrt{5}}{5}) = - \frac{32 \sqrt{5}}{125} - 8 (- \frac{2^{3} {\sqrt{5}}^{3}}{5^{3}}) + 16 (- \frac{2 \sqrt{5}}{5})$

Step 15.2.1.8

Simplify the numerator.

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Step 15.2.1.8.1

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

$f (- \frac{2 \sqrt{5}}{5}) = - \frac{32 \sqrt{5}}{125} - 8 (- \frac{8 {\sqrt{5}}^{3}}{5^{3}}) + 16 (- \frac{2 \sqrt{5}}{5})$

Step 15.2.1.8.2

Rewrite ${\sqrt{5}}^{3}$ as $\sqrt{5^{3}}$ .

$f (- \frac{2 \sqrt{5}}{5}) = - \frac{32 \sqrt{5}}{125} - 8 (- \frac{8 \sqrt{5^{3}}}{5^{3}}) + 16 (- \frac{2 \sqrt{5}}{5})$

Step 15.2.1.8.3

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

$f (- \frac{2 \sqrt{5}}{5}) = - \frac{32 \sqrt{5}}{125} - 8 (- \frac{8 \sqrt{125}}{5^{3}}) + 16 (- \frac{2 \sqrt{5}}{5})$

Step 15.2.1.8.4

Rewrite $125$ as $5^{2} \cdot 5$ .

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Step 15.2.1.8.4.1

Factor $25$ out of $125$ .

$f (- \frac{2 \sqrt{5}}{5}) = - \frac{32 \sqrt{5}}{125} - 8 (- \frac{8 \sqrt{25 (5)}}{5^{3}}) + 16 (- \frac{2 \sqrt{5}}{5})$

Step 15.2.1.8.4.2

Rewrite $25$ as $5^{2}$ .

$f (- \frac{2 \sqrt{5}}{5}) = - \frac{32 \sqrt{5}}{125} - 8 (- \frac{8 \sqrt{5^{2} \cdot 5}}{5^{3}}) + 16 (- \frac{2 \sqrt{5}}{5})$

Step 15.2.1.8.5

Pull terms out from under the radical.

$f (- \frac{2 \sqrt{5}}{5}) = - \frac{32 \sqrt{5}}{125} - 8 (- \frac{8 \cdot (5 \sqrt{5})}{5^{3}}) + 16 (- \frac{2 \sqrt{5}}{5})$

Step 15.2.1.8.6

Multiply $8$ by $5$ .

$f (- \frac{2 \sqrt{5}}{5}) = - \frac{32 \sqrt{5}}{125} - 8 (- \frac{40 \sqrt{5}}{5^{3}}) + 16 (- \frac{2 \sqrt{5}}{5})$

Step 15.2.1.9

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

$f (- \frac{2 \sqrt{5}}{5}) = - \frac{32 \sqrt{5}}{125} - 8 (- \frac{40 \sqrt{5}}{125}) + 16 (- \frac{2 \sqrt{5}}{5})$

Step 15.2.1.10

Cancel the common factor of $40$ and $125$ .

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Step 15.2.1.10.1

Factor $5$ out of $40 \sqrt{5}$ .

$f (- \frac{2 \sqrt{5}}{5}) = - \frac{32 \sqrt{5}}{125} - 8 (- \frac{5 (8 \sqrt{5})}{125}) + 16 (- \frac{2 \sqrt{5}}{5})$

Step 15.2.1.10.2

Cancel the common factors.

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Step 15.2.1.10.2.1

Factor $5$ out of $125$ .

$f (- \frac{2 \sqrt{5}}{5}) = - \frac{32 \sqrt{5}}{125} - 8 (- \frac{5 (8 \sqrt{5})}{5 (25)}) + 16 (- \frac{2 \sqrt{5}}{5})$

Step 15.2.1.10.2.2

Cancel the common factor.

$f (- \frac{2 \sqrt{5}}{5}) = - \frac{32 \sqrt{5}}{125} - 8 (- \frac{5 (8 \sqrt{5})}{5 \cdot 25}) + 16 (- \frac{2 \sqrt{5}}{5})$

Step 15.2.1.10.2.3

Rewrite the expression.

$f (- \frac{2 \sqrt{5}}{5}) = - \frac{32 \sqrt{5}}{125} - 8 (- \frac{8 \sqrt{5}}{25}) + 16 (- \frac{2 \sqrt{5}}{5})$

Step 15.2.1.11

Multiply $- 8 (- \frac{8 \sqrt{5}}{25})$ .

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Step 15.2.1.11.1

Multiply $- 1$ by $- 8$ .

$f (- \frac{2 \sqrt{5}}{5}) = - \frac{32 \sqrt{5}}{125} + 8 (\frac{8 \sqrt{5}}{25}) + 16 (- \frac{2 \sqrt{5}}{5})$

Step 15.2.1.11.2

Combine $8$ and $\frac{8 \sqrt{5}}{25}$ .

$f (- \frac{2 \sqrt{5}}{5}) = - \frac{32 \sqrt{5}}{125} + \frac{8 (8 \sqrt{5})}{25} + 16 (- \frac{2 \sqrt{5}}{5})$

Step 15.2.1.11.3

Multiply $8$ by $8$ .

$f (- \frac{2 \sqrt{5}}{5}) = - \frac{32 \sqrt{5}}{125} + \frac{64 \sqrt{5}}{25} + 16 (- \frac{2 \sqrt{5}}{5})$

Step 15.2.1.12

Multiply $16 (- \frac{2 \sqrt{5}}{5})$ .

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Step 15.2.1.12.1

Multiply $- 1$ by $16$ .

$f (- \frac{2 \sqrt{5}}{5}) = - \frac{32 \sqrt{5}}{125} + \frac{64 \sqrt{5}}{25} - 16 \frac{2 \sqrt{5}}{5}$

Step 15.2.1.12.2

Combine $- 16$ and $\frac{2 \sqrt{5}}{5}$ .

$f (- \frac{2 \sqrt{5}}{5}) = - \frac{32 \sqrt{5}}{125} + \frac{64 \sqrt{5}}{25} + \frac{- 16 (2 \sqrt{5})}{5}$

Step 15.2.1.12.3

Multiply $2$ by $- 16$ .

$f (- \frac{2 \sqrt{5}}{5}) = - \frac{32 \sqrt{5}}{125} + \frac{64 \sqrt{5}}{25} + \frac{- 32 \sqrt{5}}{5}$

Step 15.2.1.13

Move the negative in front of the fraction.

$f (- \frac{2 \sqrt{5}}{5}) = - \frac{32 \sqrt{5}}{125} + \frac{64 \sqrt{5}}{25} - \frac{32 \sqrt{5}}{5}$

Step 15.2.2

Find the common denominator.

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Step 15.2.2.1

Multiply $\frac{64 \sqrt{5}}{25}$ by $\frac{5}{5}$ .

$f (- \frac{2 \sqrt{5}}{5}) = - \frac{32 \sqrt{5}}{125} + \frac{64 \sqrt{5}}{25} \cdot \frac{5}{5} - \frac{32 \sqrt{5}}{5}$

Step 15.2.2.2

Multiply $\frac{64 \sqrt{5}}{25}$ by $\frac{5}{5}$ .

$f (- \frac{2 \sqrt{5}}{5}) = - \frac{32 \sqrt{5}}{125} + \frac{64 \sqrt{5} \cdot 5}{25 \cdot 5} - \frac{32 \sqrt{5}}{5}$

Step 15.2.2.3

Multiply $\frac{32 \sqrt{5}}{5}$ by $\frac{25}{25}$ .

$f (- \frac{2 \sqrt{5}}{5}) = - \frac{32 \sqrt{5}}{125} + \frac{64 \sqrt{5} \cdot 5}{25 \cdot 5} - (\frac{32 \sqrt{5}}{5} \cdot \frac{25}{25})$

Step 15.2.2.4

Multiply $\frac{32 \sqrt{5}}{5}$ by $\frac{25}{25}$ .

$f (- \frac{2 \sqrt{5}}{5}) = - \frac{32 \sqrt{5}}{125} + \frac{64 \sqrt{5} \cdot 5}{25 \cdot 5} - \frac{32 \sqrt{5} \cdot 25}{5 \cdot 25}$

Step 15.2.2.5

Reorder the factors of $25 \cdot 5$ .

$f (- \frac{2 \sqrt{5}}{5}) = - \frac{32 \sqrt{5}}{125} + \frac{64 \sqrt{5} \cdot 5}{5 \cdot 25} - \frac{32 \sqrt{5} \cdot 25}{5 \cdot 25}$

Step 15.2.2.6

Multiply $5$ by $25$ .

$f (- \frac{2 \sqrt{5}}{5}) = - \frac{32 \sqrt{5}}{125} + \frac{64 \sqrt{5} \cdot 5}{125} - \frac{32 \sqrt{5} \cdot 25}{5 \cdot 25}$

Step 15.2.2.7

Multiply $5$ by $25$ .

$f (- \frac{2 \sqrt{5}}{5}) = - \frac{32 \sqrt{5}}{125} + \frac{64 \sqrt{5} \cdot 5}{125} - \frac{32 \sqrt{5} \cdot 25}{125}$

Step 15.2.3

Combine the numerators over the common denominator.

$f (- \frac{2 \sqrt{5}}{5}) = \frac{- 32 \sqrt{5} + 64 \sqrt{5} \cdot 5 - 32 \sqrt{5} \cdot 25}{125}$

Step 15.2.4

Simplify each term.

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Step 15.2.4.1

Multiply $5$ by $64$ .

$f (- \frac{2 \sqrt{5}}{5}) = \frac{- 32 \sqrt{5} + 320 \sqrt{5} - 32 \sqrt{5} \cdot 25}{125}$

Step 15.2.4.2

Multiply $25$ by $- 32$ .

$f (- \frac{2 \sqrt{5}}{5}) = \frac{- 32 \sqrt{5} + 320 \sqrt{5} - 800 \sqrt{5}}{125}$

Step 15.2.5

Simplify by adding terms.

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Step 15.2.5.1

Add $- 32 \sqrt{5}$ and $320 \sqrt{5}$ .

$f (- \frac{2 \sqrt{5}}{5}) = \frac{288 \sqrt{5} - 800 \sqrt{5}}{125}$

Step 15.2.5.2

Subtract $800 \sqrt{5}$ from $288 \sqrt{5}$ .

$f (- \frac{2 \sqrt{5}}{5}) = \frac{- 512 \sqrt{5}}{125}$

Step 15.2.5.3

Move the negative in front of the fraction.

$f (- \frac{2 \sqrt{5}}{5}) = - \frac{512 \sqrt{5}}{125}$

Step 15.2.6

The final answer is $- \frac{512 \sqrt{5}}{125}$ .

$y = - \frac{512 \sqrt{5}}{125}$

Step 16

Evaluate the second derivative at $x = 2$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$20 {(2)}^{3} - 48 \cdot 2$

Step 17

Evaluate the second derivative.

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Step 17.1

Simplify each term.

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Step 17.1.1

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

$20 \cdot 8 - 48 \cdot 2$

Step 17.1.2

Multiply $20$ by $8$ .

$160 - 48 \cdot 2$

Step 17.1.3

Multiply $- 48$ by $2$ .

$160 - 96$

Step 17.2

Subtract $96$ from $160$ .

$64$

Step 18

$x = 2$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = 2$ is a local minimum

Step 19

Find the y-value when $x = 2$ .

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Step 19.1

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

$f (2) = {(2)}^{5} - 8 {(2)}^{3} + 16 (2)$

Step 19.2

Simplify the result.

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Step 19.2.1

Simplify each term.

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Step 19.2.1.1

Raise $2$ to the power of $5$ .

$f (2) = 32 - 8 {(2)}^{3} + 16 (2)$

Step 19.2.1.2

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

$f (2) = 32 - 8 \cdot 8 + 16 (2)$

Step 19.2.1.3

Multiply $- 8$ by $8$ .

$f (2) = 32 - 64 + 16 (2)$

Step 19.2.1.4

Multiply $16$ by $2$ .

$f (2) = 32 - 64 + 32$

Step 19.2.2

Simplify by adding and subtracting.

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Step 19.2.2.1

Subtract $64$ from $32$ .

$f (2) = - 32 + 32$

Step 19.2.2.2

Add $- 32$ and $32$ .

$f (2) = 0$

Step 19.2.3

The final answer is $0$ .

$y = 0$

Step 20

Evaluate the second derivative at $x = - 2$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$20 {(- 2)}^{3} - 48 \cdot - 2$

Step 21

Evaluate the second derivative.

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Step 21.1

Simplify each term.

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Step 21.1.1

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

$20 \cdot - 8 - 48 \cdot - 2$

Step 21.1.2

Multiply $20$ by $- 8$ .

$- 160 - 48 \cdot - 2$

Step 21.1.3

Multiply $- 48$ by $- 2$ .

$- 160 + 96$

Step 21.2

Add $- 160$ and $96$ .

$- 64$

Step 22

$x = - 2$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = - 2$ is a local maximum

Step 23

Find the y-value when $x = - 2$ .

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Step 23.1

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

$f (- 2) = {(- 2)}^{5} - 8 {(- 2)}^{3} + 16 (- 2)$

Step 23.2

Simplify the result.

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Step 23.2.1

Simplify each term.

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Step 23.2.1.1

Raise $- 2$ to the power of $5$ .

$f (- 2) = - 32 - 8 {(- 2)}^{3} + 16 (- 2)$

Step 23.2.1.2

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

$f (- 2) = - 32 - 8 \cdot - 8 + 16 (- 2)$

Step 23.2.1.3

Multiply $- 8$ by $- 8$ .

$f (- 2) = - 32 + 64 + 16 (- 2)$

Step 23.2.1.4

Multiply $16$ by $- 2$ .

$f (- 2) = - 32 + 64 - 32$

Step 23.2.2

Simplify by adding and subtracting.

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Step 23.2.2.1

Add $- 32$ and $64$ .

$f (- 2) = 32 - 32$

Step 23.2.2.2

Subtract $32$ from $32$ .

$f (- 2) = 0$

Step 23.2.3

The final answer is $0$ .

$y = 0$

Step 24

These are the local extrema for $f (x) = x^{5} - 8 x^{3} + 16 x$ .

$(\frac{2 \sqrt{5}}{5}, \frac{512 \sqrt{5}}{125})$ is a local maxima

$(- \frac{2 \sqrt{5}}{5}, - \frac{512 \sqrt{5}}{125})$ is a local minima

$(2, 0)$ is a local minima

$(- 2, 0)$ is a local maxima

Step 25