Calculus Examples

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

Step 1

Find the second derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Differentiate.

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Step 1.1.1.1

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 1.1.1.2

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 1.1.2

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

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Step 1.1.2.1

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 1.1.2.2

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 1.1.2.3

Multiply $3$ by $- 8$ .

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

Step 1.1.3

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

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Step 1.1.3.1

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 1.1.3.2

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 1.1.3.3

Multiply $16$ by $1$ .

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

Step 1.2

Find the second derivative.

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Step 1.2.1

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

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

Step 1.2.2

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

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Step 1.2.2.1

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

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

Step 1.2.2.2

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

$5 (4 x^{3}) + \frac{d}{d x} [- 24 x^{2}] + \frac{d}{d x} [16]$

Step 1.2.2.3

Multiply $4$ by $5$ .

$20 x^{3} + \frac{d}{d x} [- 24 x^{2}] + \frac{d}{d x} [16]$

Step 1.2.3

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

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Step 1.2.3.1

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

$20 x^{3} - 24 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [16]$

Step 1.2.3.2

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

$20 x^{3} - 24 (2 x) + \frac{d}{d x} [16]$

Step 1.2.3.3

Multiply $2$ by $- 24$ .

$20 x^{3} - 48 x + \frac{d}{d x} [16]$

Step 1.2.4

Differentiate using the Constant Rule.

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Step 1.2.4.1

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

$20 x^{3} - 48 x + 0$

Step 1.2.4.2

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

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

Step 1.3

The second derivative of $f (x)$ with respect to $x$ is $20 x^{3} - 48 x$ .

$20 x^{3} - 48 x$

Step 2

Set the second derivative equal to $0$ then solve the equation $20 x^{3} - 48 x = 0$ .

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Step 2.1

Set the second derivative equal to $0$ .

$20 x^{3} - 48 x = 0$

Step 2.2

Factor $4 x$ out of $20 x^{3} - 48 x$ .

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Step 2.2.1

Factor $4 x$ out of $20 x^{3}$ .

$4 x (5 x^{2}) - 48 x = 0$

Step 2.2.2

Factor $4 x$ out of $- 48 x$ .

$4 x (5 x^{2}) + 4 x (- 12) = 0$

Step 2.2.3

Factor $4 x$ out of $4 x (5 x^{2}) + 4 x (- 12)$ .

$4 x (5 x^{2} - 12) = 0$

Step 2.3

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

$x = 0$

$5 x^{2} - 12 = 0$

Step 2.4

Set $x$ equal to $0$ .

$x = 0$

Step 2.5

Set $5 x^{2} - 12$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $5 x^{2} - 12$ equal to $0$ .

$5 x^{2} - 12 = 0$

Step 2.5.2

Solve $5 x^{2} - 12 = 0$ for $x$ .

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Step 2.5.2.1

Add $12$ to both sides of the equation.

$5 x^{2} = 12$

Step 2.5.2.2

Divide each term in $5 x^{2} = 12$ by $5$ and simplify.

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Step 2.5.2.2.1

Divide each term in $5 x^{2} = 12$ by $5$ .

$\frac{5 x^{2}}{5} = \frac{12}{5}$

Step 2.5.2.2.2

Simplify the left side.

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Step 2.5.2.2.2.1

Cancel the common factor of $5$ .

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Step 2.5.2.2.2.1.1

Cancel the common factor.

$\frac{5 x^{2}}{5} = \frac{12}{5}$

Step 2.5.2.2.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{12}{5}$

Step 2.5.2.3

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

$x = \pm \sqrt{\frac{12}{5}}$

Step 2.5.2.4

Simplify $\pm \sqrt{\frac{12}{5}}$ .

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Step 2.5.2.4.1

Rewrite $\sqrt{\frac{12}{5}}$ as $\frac{\sqrt{12}}{\sqrt{5}}$ .

$x = \pm \frac{\sqrt{12}}{\sqrt{5}}$

Step 2.5.2.4.2

Simplify the numerator.

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Step 2.5.2.4.2.1

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 2.5.2.4.2.1.1

Factor $4$ out of $12$ .

$x = \pm \frac{\sqrt{4 (3)}}{\sqrt{5}}$

Step 2.5.2.4.2.1.2

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

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

Step 2.5.2.4.2.2

Pull terms out from under the radical.

$x = \pm \frac{2 \sqrt{3}}{\sqrt{5}}$

Step 2.5.2.4.3

Multiply $\frac{2 \sqrt{3}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$x = \pm \frac{2 \sqrt{3}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Step 2.5.2.4.4

Combine and simplify the denominator.

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Step 2.5.2.4.4.1

Multiply $\frac{2 \sqrt{3}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$x = \pm \frac{2 \sqrt{3} \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 2.5.2.4.4.2

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

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

Step 2.5.2.4.4.3

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

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

Step 2.5.2.4.4.4

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

$x = \pm \frac{2 \sqrt{3} \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Step 2.5.2.4.4.5

Add $1$ and $1$ .

$x = \pm \frac{2 \sqrt{3} \sqrt{5}}{{\sqrt{5}}^{2}}$

Step 2.5.2.4.4.6

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

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Step 2.5.2.4.4.6.1

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

$x = \pm \frac{2 \sqrt{3} \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Step 2.5.2.4.4.6.2

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

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

Step 2.5.2.4.4.6.3

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

$x = \pm \frac{2 \sqrt{3} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 2.5.2.4.4.6.4

Cancel the common factor of $2$ .

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Step 2.5.2.4.4.6.4.1

Cancel the common factor.

$x = \pm \frac{2 \sqrt{3} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 2.5.2.4.4.6.4.2

Rewrite the expression.

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

Step 2.5.2.4.4.6.5

Evaluate the exponent.

$x = \pm \frac{2 \sqrt{3} \sqrt{5}}{5}$

Step 2.5.2.4.5

Simplify the numerator.

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Step 2.5.2.4.5.1

Combine using the product rule for radicals.

$x = \pm \frac{2 \sqrt{5 \cdot 3}}{5}$

Step 2.5.2.4.5.2

Multiply $5$ by $3$ .

$x = \pm \frac{2 \sqrt{15}}{5}$

Step 2.5.2.5

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

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Step 2.5.2.5.1

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

$x = \frac{2 \sqrt{15}}{5}$

Step 2.5.2.5.2

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

$x = - \frac{2 \sqrt{15}}{5}$

Step 2.5.2.5.3

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

$x = \frac{2 \sqrt{15}}{5}, - \frac{2 \sqrt{15}}{5}$

Step 2.6

The final solution is all the values that make $4 x (5 x^{2} - 12) = 0$ true.

$x = 0, \frac{2 \sqrt{15}}{5}, - \frac{2 \sqrt{15}}{5}$

Step 3

Find the points where the second derivative is $0$ .

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Step 3.1

Substitute $0$ in $f (x) = x^{5} - 8 x^{3} + 16 x$ to find the value of $y$ .

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Step 3.1.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = {(0)}^{5} - 8 {(0)}^{3} + 16 (0)$

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Simplify each term.

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Step 3.1.2.1.1

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

$f (0) = 0 - 8 {(0)}^{3} + 16 (0)$

Step 3.1.2.1.2

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

$f (0) = 0 - 8 \cdot 0 + 16 (0)$

Step 3.1.2.1.3

Multiply $- 8$ by $0$ .

$f (0) = 0 + 0 + 16 (0)$

Step 3.1.2.1.4

Multiply $16$ by $0$ .

$f (0) = 0 + 0 + 0$

Step 3.1.2.2

Simplify by adding numbers.

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Step 3.1.2.2.1

Add $0$ and $0$ .

$f (0) = 0 + 0$

Step 3.1.2.2.2

Add $0$ and $0$ .

$f (0) = 0$

Step 3.1.2.3

The final answer is $0$ .

$0$

Step 3.2

The point found by substituting $0$ in $f (x) = x^{5} - 8 x^{3} + 16 x$ is $(0, 0)$ . This point can be an inflection point.

$(0, 0)$

Step 3.3

Substitute $\frac{2 \sqrt{15}}{5}$ in $f (x) = x^{5} - 8 x^{3} + 16 x$ to find the value of $y$ .

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Step 3.3.1

Replace the variable $x$ with $\frac{2 \sqrt{15}}{5}$ in the expression.

$f (\frac{2 \sqrt{15}}{5}) = {(\frac{2 \sqrt{15}}{5})}^{5} - 8 {(\frac{2 \sqrt{15}}{5})}^{3} + 16 (\frac{2 \sqrt{15}}{5})$

Step 3.3.2

Simplify the result.

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Step 3.3.2.1

Simplify each term.

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Step 3.3.2.1.1

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

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Step 3.3.2.1.1.1

Apply the product rule to $\frac{2 \sqrt{15}}{5}$ .

$f (\frac{2 \sqrt{15}}{5}) = \frac{{(2 \sqrt{15})}^{5}}{5^{5}} - 8 {(\frac{2 \sqrt{15}}{5})}^{3} + 16 (\frac{2 \sqrt{15}}{5})$

Step 3.3.2.1.1.2

Apply the product rule to $2 \sqrt{15}$ .

$f (\frac{2 \sqrt{15}}{5}) = \frac{2^{5} {\sqrt{15}}^{5}}{5^{5}} - 8 {(\frac{2 \sqrt{15}}{5})}^{3} + 16 (\frac{2 \sqrt{15}}{5})$

Step 3.3.2.1.2

Simplify the numerator.

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Step 3.3.2.1.2.1

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

$f (\frac{2 \sqrt{15}}{5}) = \frac{32 {\sqrt{15}}^{5}}{5^{5}} - 8 {(\frac{2 \sqrt{15}}{5})}^{3} + 16 (\frac{2 \sqrt{15}}{5})$

Step 3.3.2.1.2.2

Rewrite ${\sqrt{15}}^{5}$ as $\sqrt{15^{5}}$ .

$f (\frac{2 \sqrt{15}}{5}) = \frac{32 \sqrt{15^{5}}}{5^{5}} - 8 {(\frac{2 \sqrt{15}}{5})}^{3} + 16 (\frac{2 \sqrt{15}}{5})$

Step 3.3.2.1.2.3

Raise $15$ to the power of $5$ .

$f (\frac{2 \sqrt{15}}{5}) = \frac{32 \sqrt{759375}}{5^{5}} - 8 {(\frac{2 \sqrt{15}}{5})}^{3} + 16 (\frac{2 \sqrt{15}}{5})$

Step 3.3.2.1.2.4

Rewrite $759375$ as $225^{2} \cdot 15$ .

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Step 3.3.2.1.2.4.1

Factor $50625$ out of $759375$ .

$f (\frac{2 \sqrt{15}}{5}) = \frac{32 \sqrt{50625 (15)}}{5^{5}} - 8 {(\frac{2 \sqrt{15}}{5})}^{3} + 16 (\frac{2 \sqrt{15}}{5})$

Step 3.3.2.1.2.4.2

Rewrite $50625$ as $225^{2}$ .

$f (\frac{2 \sqrt{15}}{5}) = \frac{32 \sqrt{225^{2} \cdot 15}}{5^{5}} - 8 {(\frac{2 \sqrt{15}}{5})}^{3} + 16 (\frac{2 \sqrt{15}}{5})$

Step 3.3.2.1.2.5

Pull terms out from under the radical.

$f (\frac{2 \sqrt{15}}{5}) = \frac{32 \cdot (225 \sqrt{15})}{5^{5}} - 8 {(\frac{2 \sqrt{15}}{5})}^{3} + 16 (\frac{2 \sqrt{15}}{5})$

Step 3.3.2.1.2.6

Multiply $32$ by $225$ .

$f (\frac{2 \sqrt{15}}{5}) = \frac{7200 \sqrt{15}}{5^{5}} - 8 {(\frac{2 \sqrt{15}}{5})}^{3} + 16 (\frac{2 \sqrt{15}}{5})$

Step 3.3.2.1.3

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

$f (\frac{2 \sqrt{15}}{5}) = \frac{7200 \sqrt{15}}{3125} - 8 {(\frac{2 \sqrt{15}}{5})}^{3} + 16 (\frac{2 \sqrt{15}}{5})$

Step 3.3.2.1.4

Cancel the common factor of $7200$ and $3125$ .

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Step 3.3.2.1.4.1

Factor $25$ out of $7200 \sqrt{15}$ .

$f (\frac{2 \sqrt{15}}{5}) = \frac{25 (288 \sqrt{15})}{3125} - 8 {(\frac{2 \sqrt{15}}{5})}^{3} + 16 (\frac{2 \sqrt{15}}{5})$

Step 3.3.2.1.4.2

Cancel the common factors.

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Step 3.3.2.1.4.2.1

Factor $25$ out of $3125$ .

$f (\frac{2 \sqrt{15}}{5}) = \frac{25 (288 \sqrt{15})}{25 (125)} - 8 {(\frac{2 \sqrt{15}}{5})}^{3} + 16 (\frac{2 \sqrt{15}}{5})$

Step 3.3.2.1.4.2.2

Cancel the common factor.

$f (\frac{2 \sqrt{15}}{5}) = \frac{25 (288 \sqrt{15})}{25 \cdot 125} - 8 {(\frac{2 \sqrt{15}}{5})}^{3} + 16 (\frac{2 \sqrt{15}}{5})$

Step 3.3.2.1.4.2.3

Rewrite the expression.

$f (\frac{2 \sqrt{15}}{5}) = \frac{288 \sqrt{15}}{125} - 8 {(\frac{2 \sqrt{15}}{5})}^{3} + 16 (\frac{2 \sqrt{15}}{5})$

Step 3.3.2.1.5

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

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Step 3.3.2.1.5.1

Apply the product rule to $\frac{2 \sqrt{15}}{5}$ .

$f (\frac{2 \sqrt{15}}{5}) = \frac{288 \sqrt{15}}{125} - 8 \frac{{(2 \sqrt{15})}^{3}}{5^{3}} + 16 (\frac{2 \sqrt{15}}{5})$

Step 3.3.2.1.5.2

Apply the product rule to $2 \sqrt{15}$ .

$f (\frac{2 \sqrt{15}}{5}) = \frac{288 \sqrt{15}}{125} - 8 \frac{2^{3} {\sqrt{15}}^{3}}{5^{3}} + 16 (\frac{2 \sqrt{15}}{5})$

Step 3.3.2.1.6

Simplify the numerator.

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Step 3.3.2.1.6.1

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

$f (\frac{2 \sqrt{15}}{5}) = \frac{288 \sqrt{15}}{125} - 8 \frac{8 {\sqrt{15}}^{3}}{5^{3}} + 16 (\frac{2 \sqrt{15}}{5})$

Step 3.3.2.1.6.2

Rewrite ${\sqrt{15}}^{3}$ as $\sqrt{15^{3}}$ .

$f (\frac{2 \sqrt{15}}{5}) = \frac{288 \sqrt{15}}{125} - 8 \frac{8 \sqrt{15^{3}}}{5^{3}} + 16 (\frac{2 \sqrt{15}}{5})$

Step 3.3.2.1.6.3

Raise $15$ to the power of $3$ .

$f (\frac{2 \sqrt{15}}{5}) = \frac{288 \sqrt{15}}{125} - 8 \frac{8 \sqrt{3375}}{5^{3}} + 16 (\frac{2 \sqrt{15}}{5})$

Step 3.3.2.1.6.4

Rewrite $3375$ as $15^{2} \cdot 15$ .

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Step 3.3.2.1.6.4.1

Factor $225$ out of $3375$ .

$f (\frac{2 \sqrt{15}}{5}) = \frac{288 \sqrt{15}}{125} - 8 \frac{8 \sqrt{225 (15)}}{5^{3}} + 16 (\frac{2 \sqrt{15}}{5})$

Step 3.3.2.1.6.4.2

Rewrite $225$ as $15^{2}$ .

$f (\frac{2 \sqrt{15}}{5}) = \frac{288 \sqrt{15}}{125} - 8 \frac{8 \sqrt{15^{2} \cdot 15}}{5^{3}} + 16 (\frac{2 \sqrt{15}}{5})$

Step 3.3.2.1.6.5

Pull terms out from under the radical.

$f (\frac{2 \sqrt{15}}{5}) = \frac{288 \sqrt{15}}{125} - 8 \frac{8 \cdot (15 \sqrt{15})}{5^{3}} + 16 (\frac{2 \sqrt{15}}{5})$

Step 3.3.2.1.6.6

Multiply $8$ by $15$ .

$f (\frac{2 \sqrt{15}}{5}) = \frac{288 \sqrt{15}}{125} - 8 \frac{120 \sqrt{15}}{5^{3}} + 16 (\frac{2 \sqrt{15}}{5})$

Step 3.3.2.1.7

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

$f (\frac{2 \sqrt{15}}{5}) = \frac{288 \sqrt{15}}{125} - 8 \frac{120 \sqrt{15}}{125} + 16 (\frac{2 \sqrt{15}}{5})$

Step 3.3.2.1.8

Cancel the common factor of $120$ and $125$ .

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Step 3.3.2.1.8.1

Factor $5$ out of $120 \sqrt{15}$ .

$f (\frac{2 \sqrt{15}}{5}) = \frac{288 \sqrt{15}}{125} - 8 \frac{5 (24 \sqrt{15})}{125} + 16 (\frac{2 \sqrt{15}}{5})$

Step 3.3.2.1.8.2

Cancel the common factors.

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Step 3.3.2.1.8.2.1

Factor $5$ out of $125$ .

$f (\frac{2 \sqrt{15}}{5}) = \frac{288 \sqrt{15}}{125} - 8 \frac{5 (24 \sqrt{15})}{5 (25)} + 16 (\frac{2 \sqrt{15}}{5})$

Step 3.3.2.1.8.2.2

Cancel the common factor.

$f (\frac{2 \sqrt{15}}{5}) = \frac{288 \sqrt{15}}{125} - 8 \frac{5 (24 \sqrt{15})}{5 \cdot 25} + 16 (\frac{2 \sqrt{15}}{5})$

Step 3.3.2.1.8.2.3

Rewrite the expression.

$f (\frac{2 \sqrt{15}}{5}) = \frac{288 \sqrt{15}}{125} - 8 \frac{24 \sqrt{15}}{25} + 16 (\frac{2 \sqrt{15}}{5})$

Step 3.3.2.1.9

Multiply $- 8 \frac{24 \sqrt{15}}{25}$ .

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Step 3.3.2.1.9.1

Combine $- 8$ and $\frac{24 \sqrt{15}}{25}$ .

$f (\frac{2 \sqrt{15}}{5}) = \frac{288 \sqrt{15}}{125} + \frac{- 8 (24 \sqrt{15})}{25} + 16 (\frac{2 \sqrt{15}}{5})$

Step 3.3.2.1.9.2

Multiply $24$ by $- 8$ .

$f (\frac{2 \sqrt{15}}{5}) = \frac{288 \sqrt{15}}{125} + \frac{- 192 \sqrt{15}}{25} + 16 (\frac{2 \sqrt{15}}{5})$

Step 3.3.2.1.10

Move the negative in front of the fraction.

$f (\frac{2 \sqrt{15}}{5}) = \frac{288 \sqrt{15}}{125} - \frac{192 \sqrt{15}}{25} + 16 (\frac{2 \sqrt{15}}{5})$

Step 3.3.2.1.11

Multiply $16 (\frac{2 \sqrt{15}}{5})$ .

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Step 3.3.2.1.11.1

Combine $16$ and $\frac{2 \sqrt{15}}{5}$ .

$f (\frac{2 \sqrt{15}}{5}) = \frac{288 \sqrt{15}}{125} - \frac{192 \sqrt{15}}{25} + \frac{16 (2 \sqrt{15})}{5}$

Step 3.3.2.1.11.2

Multiply $2$ by $16$ .

$f (\frac{2 \sqrt{15}}{5}) = \frac{288 \sqrt{15}}{125} - \frac{192 \sqrt{15}}{25} + \frac{32 \sqrt{15}}{5}$

Step 3.3.2.2

Find the common denominator.

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Step 3.3.2.2.1

Multiply $\frac{192 \sqrt{15}}{25}$ by $\frac{5}{5}$ .

$f (\frac{2 \sqrt{15}}{5}) = \frac{288 \sqrt{15}}{125} - (\frac{192 \sqrt{15}}{25} \cdot \frac{5}{5}) + \frac{32 \sqrt{15}}{5}$

Step 3.3.2.2.2

Multiply $\frac{192 \sqrt{15}}{25}$ by $\frac{5}{5}$ .

$f (\frac{2 \sqrt{15}}{5}) = \frac{288 \sqrt{15}}{125} - \frac{192 \sqrt{15} \cdot 5}{25 \cdot 5} + \frac{32 \sqrt{15}}{5}$

Step 3.3.2.2.3

Multiply $\frac{32 \sqrt{15}}{5}$ by $\frac{25}{25}$ .

$f (\frac{2 \sqrt{15}}{5}) = \frac{288 \sqrt{15}}{125} - \frac{192 \sqrt{15} \cdot 5}{25 \cdot 5} + \frac{32 \sqrt{15}}{5} \cdot \frac{25}{25}$

Step 3.3.2.2.4

Multiply $\frac{32 \sqrt{15}}{5}$ by $\frac{25}{25}$ .

$f (\frac{2 \sqrt{15}}{5}) = \frac{288 \sqrt{15}}{125} - \frac{192 \sqrt{15} \cdot 5}{25 \cdot 5} + \frac{32 \sqrt{15} \cdot 25}{5 \cdot 25}$

Step 3.3.2.2.5

Reorder the factors of $25 \cdot 5$ .

$f (\frac{2 \sqrt{15}}{5}) = \frac{288 \sqrt{15}}{125} - \frac{192 \sqrt{15} \cdot 5}{5 \cdot 25} + \frac{32 \sqrt{15} \cdot 25}{5 \cdot 25}$

Step 3.3.2.2.6

Multiply $5$ by $25$ .

$f (\frac{2 \sqrt{15}}{5}) = \frac{288 \sqrt{15}}{125} - \frac{192 \sqrt{15} \cdot 5}{125} + \frac{32 \sqrt{15} \cdot 25}{5 \cdot 25}$

Step 3.3.2.2.7

Multiply $5$ by $25$ .

$f (\frac{2 \sqrt{15}}{5}) = \frac{288 \sqrt{15}}{125} - \frac{192 \sqrt{15} \cdot 5}{125} + \frac{32 \sqrt{15} \cdot 25}{125}$

Step 3.3.2.3

Combine the numerators over the common denominator.

$f (\frac{2 \sqrt{15}}{5}) = \frac{288 \sqrt{15} - 192 \sqrt{15} \cdot 5 + 32 \sqrt{15} \cdot 25}{125}$

Step 3.3.2.4

Simplify each term.

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Step 3.3.2.4.1

Multiply $5$ by $- 192$ .

$f (\frac{2 \sqrt{15}}{5}) = \frac{288 \sqrt{15} - 960 \sqrt{15} + 32 \sqrt{15} \cdot 25}{125}$

Step 3.3.2.4.2

Multiply $25$ by $32$ .

$f (\frac{2 \sqrt{15}}{5}) = \frac{288 \sqrt{15} - 960 \sqrt{15} + 800 \sqrt{15}}{125}$

Step 3.3.2.5

Simplify by adding terms.

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Step 3.3.2.5.1

Subtract $960 \sqrt{15}$ from $288 \sqrt{15}$ .

$f (\frac{2 \sqrt{15}}{5}) = \frac{- 672 \sqrt{15} + 800 \sqrt{15}}{125}$

Step 3.3.2.5.2

Add $- 672 \sqrt{15}$ and $800 \sqrt{15}$ .

$f (\frac{2 \sqrt{15}}{5}) = \frac{128 \sqrt{15}}{125}$

Step 3.3.2.6

The final answer is $\frac{128 \sqrt{15}}{125}$ .

$\frac{128 \sqrt{15}}{125}$

Step 3.4

The point found by substituting $\frac{2 \sqrt{15}}{5}$ in $f (x) = x^{5} - 8 x^{3} + 16 x$ is $(\frac{2 \sqrt{15}}{5}, \frac{128 \sqrt{15}}{125})$ . This point can be an inflection point.

$(\frac{2 \sqrt{15}}{5}, \frac{128 \sqrt{15}}{125})$

Step 3.5

Substitute $- \frac{2 \sqrt{15}}{5}$ in $f (x) = x^{5} - 8 x^{3} + 16 x$ to find the value of $y$ .

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Step 3.5.1

Replace the variable $x$ with $- \frac{2 \sqrt{15}}{5}$ in the expression.

$f (- \frac{2 \sqrt{15}}{5}) = {(- \frac{2 \sqrt{15}}{5})}^{5} - 8 {(- \frac{2 \sqrt{15}}{5})}^{3} + 16 (- \frac{2 \sqrt{15}}{5})$

Step 3.5.2

Simplify the result.

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Step 3.5.2.1

Simplify each term.

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Step 3.5.2.1.1

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

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Step 3.5.2.1.1.1

Apply the product rule to $- \frac{2 \sqrt{15}}{5}$ .

$f (- \frac{2 \sqrt{15}}{5}) = {(- 1)}^{5} {(\frac{2 \sqrt{15}}{5})}^{5} - 8 {(- \frac{2 \sqrt{15}}{5})}^{3} + 16 (- \frac{2 \sqrt{15}}{5})$

Step 3.5.2.1.1.2

Apply the product rule to $\frac{2 \sqrt{15}}{5}$ .

$f (- \frac{2 \sqrt{15}}{5}) = {(- 1)}^{5} (\frac{{(2 \sqrt{15})}^{5}}{5^{5}}) - 8 {(- \frac{2 \sqrt{15}}{5})}^{3} + 16 (- \frac{2 \sqrt{15}}{5})$

Step 3.5.2.1.1.3

Apply the product rule to $2 \sqrt{15}$ .

$f (- \frac{2 \sqrt{15}}{5}) = {(- 1)}^{5} (\frac{2^{5} {\sqrt{15}}^{5}}{5^{5}}) - 8 {(- \frac{2 \sqrt{15}}{5})}^{3} + 16 (- \frac{2 \sqrt{15}}{5})$

Step 3.5.2.1.2

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

$f (- \frac{2 \sqrt{15}}{5}) = - \frac{2^{5} {\sqrt{15}}^{5}}{5^{5}} - 8 {(- \frac{2 \sqrt{15}}{5})}^{3} + 16 (- \frac{2 \sqrt{15}}{5})$

Step 3.5.2.1.3

Simplify the numerator.

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Step 3.5.2.1.3.1

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

$f (- \frac{2 \sqrt{15}}{5}) = - \frac{32 {\sqrt{15}}^{5}}{5^{5}} - 8 {(- \frac{2 \sqrt{15}}{5})}^{3} + 16 (- \frac{2 \sqrt{15}}{5})$

Step 3.5.2.1.3.2

Rewrite ${\sqrt{15}}^{5}$ as $\sqrt{15^{5}}$ .

$f (- \frac{2 \sqrt{15}}{5}) = - \frac{32 \sqrt{15^{5}}}{5^{5}} - 8 {(- \frac{2 \sqrt{15}}{5})}^{3} + 16 (- \frac{2 \sqrt{15}}{5})$

Step 3.5.2.1.3.3

Raise $15$ to the power of $5$ .

$f (- \frac{2 \sqrt{15}}{5}) = - \frac{32 \sqrt{759375}}{5^{5}} - 8 {(- \frac{2 \sqrt{15}}{5})}^{3} + 16 (- \frac{2 \sqrt{15}}{5})$

Step 3.5.2.1.3.4

Rewrite $759375$ as $225^{2} \cdot 15$ .

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Step 3.5.2.1.3.4.1

Factor $50625$ out of $759375$ .

$f (- \frac{2 \sqrt{15}}{5}) = - \frac{32 \sqrt{50625 (15)}}{5^{5}} - 8 {(- \frac{2 \sqrt{15}}{5})}^{3} + 16 (- \frac{2 \sqrt{15}}{5})$

Step 3.5.2.1.3.4.2

Rewrite $50625$ as $225^{2}$ .

$f (- \frac{2 \sqrt{15}}{5}) = - \frac{32 \sqrt{225^{2} \cdot 15}}{5^{5}} - 8 {(- \frac{2 \sqrt{15}}{5})}^{3} + 16 (- \frac{2 \sqrt{15}}{5})$

Step 3.5.2.1.3.5

Pull terms out from under the radical.

$f (- \frac{2 \sqrt{15}}{5}) = - \frac{32 \cdot (225 \sqrt{15})}{5^{5}} - 8 {(- \frac{2 \sqrt{15}}{5})}^{3} + 16 (- \frac{2 \sqrt{15}}{5})$

Step 3.5.2.1.3.6

Multiply $32$ by $225$ .

$f (- \frac{2 \sqrt{15}}{5}) = - \frac{7200 \sqrt{15}}{5^{5}} - 8 {(- \frac{2 \sqrt{15}}{5})}^{3} + 16 (- \frac{2 \sqrt{15}}{5})$

Step 3.5.2.1.4

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

$f (- \frac{2 \sqrt{15}}{5}) = - \frac{7200 \sqrt{15}}{3125} - 8 {(- \frac{2 \sqrt{15}}{5})}^{3} + 16 (- \frac{2 \sqrt{15}}{5})$

Step 3.5.2.1.5

Cancel the common factor of $7200$ and $3125$ .

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Step 3.5.2.1.5.1

Factor $25$ out of $7200 \sqrt{15}$ .

$f (- \frac{2 \sqrt{15}}{5}) = - \frac{25 (288 \sqrt{15})}{3125} - 8 {(- \frac{2 \sqrt{15}}{5})}^{3} + 16 (- \frac{2 \sqrt{15}}{5})$

Step 3.5.2.1.5.2

Cancel the common factors.

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Step 3.5.2.1.5.2.1

Factor $25$ out of $3125$ .

$f (- \frac{2 \sqrt{15}}{5}) = - \frac{25 (288 \sqrt{15})}{25 (125)} - 8 {(- \frac{2 \sqrt{15}}{5})}^{3} + 16 (- \frac{2 \sqrt{15}}{5})$

Step 3.5.2.1.5.2.2

Cancel the common factor.

$f (- \frac{2 \sqrt{15}}{5}) = - \frac{25 (288 \sqrt{15})}{25 \cdot 125} - 8 {(- \frac{2 \sqrt{15}}{5})}^{3} + 16 (- \frac{2 \sqrt{15}}{5})$

Step 3.5.2.1.5.2.3

Rewrite the expression.

$f (- \frac{2 \sqrt{15}}{5}) = - \frac{288 \sqrt{15}}{125} - 8 {(- \frac{2 \sqrt{15}}{5})}^{3} + 16 (- \frac{2 \sqrt{15}}{5})$

Step 3.5.2.1.6

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

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Step 3.5.2.1.6.1

Apply the product rule to $- \frac{2 \sqrt{15}}{5}$ .

$f (- \frac{2 \sqrt{15}}{5}) = - \frac{288 \sqrt{15}}{125} - 8 ({(- 1)}^{3} {(\frac{2 \sqrt{15}}{5})}^{3}) + 16 (- \frac{2 \sqrt{15}}{5})$

Step 3.5.2.1.6.2

Apply the product rule to $\frac{2 \sqrt{15}}{5}$ .

$f (- \frac{2 \sqrt{15}}{5}) = - \frac{288 \sqrt{15}}{125} - 8 ({(- 1)}^{3} (\frac{{(2 \sqrt{15})}^{3}}{5^{3}})) + 16 (- \frac{2 \sqrt{15}}{5})$

Step 3.5.2.1.6.3

Apply the product rule to $2 \sqrt{15}$ .

$f (- \frac{2 \sqrt{15}}{5}) = - \frac{288 \sqrt{15}}{125} - 8 ({(- 1)}^{3} (\frac{2^{3} {\sqrt{15}}^{3}}{5^{3}})) + 16 (- \frac{2 \sqrt{15}}{5})$

Step 3.5.2.1.7

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

$f (- \frac{2 \sqrt{15}}{5}) = - \frac{288 \sqrt{15}}{125} - 8 (- \frac{2^{3} {\sqrt{15}}^{3}}{5^{3}}) + 16 (- \frac{2 \sqrt{15}}{5})$

Step 3.5.2.1.8

Simplify the numerator.

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Step 3.5.2.1.8.1

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

$f (- \frac{2 \sqrt{15}}{5}) = - \frac{288 \sqrt{15}}{125} - 8 (- \frac{8 {\sqrt{15}}^{3}}{5^{3}}) + 16 (- \frac{2 \sqrt{15}}{5})$

Step 3.5.2.1.8.2

Rewrite ${\sqrt{15}}^{3}$ as $\sqrt{15^{3}}$ .

$f (- \frac{2 \sqrt{15}}{5}) = - \frac{288 \sqrt{15}}{125} - 8 (- \frac{8 \sqrt{15^{3}}}{5^{3}}) + 16 (- \frac{2 \sqrt{15}}{5})$

Step 3.5.2.1.8.3

Raise $15$ to the power of $3$ .

$f (- \frac{2 \sqrt{15}}{5}) = - \frac{288 \sqrt{15}}{125} - 8 (- \frac{8 \sqrt{3375}}{5^{3}}) + 16 (- \frac{2 \sqrt{15}}{5})$

Step 3.5.2.1.8.4

Rewrite $3375$ as $15^{2} \cdot 15$ .

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Step 3.5.2.1.8.4.1

Factor $225$ out of $3375$ .

$f (- \frac{2 \sqrt{15}}{5}) = - \frac{288 \sqrt{15}}{125} - 8 (- \frac{8 \sqrt{225 (15)}}{5^{3}}) + 16 (- \frac{2 \sqrt{15}}{5})$

Step 3.5.2.1.8.4.2

Rewrite $225$ as $15^{2}$ .

$f (- \frac{2 \sqrt{15}}{5}) = - \frac{288 \sqrt{15}}{125} - 8 (- \frac{8 \sqrt{15^{2} \cdot 15}}{5^{3}}) + 16 (- \frac{2 \sqrt{15}}{5})$

Step 3.5.2.1.8.5

Pull terms out from under the radical.

$f (- \frac{2 \sqrt{15}}{5}) = - \frac{288 \sqrt{15}}{125} - 8 (- \frac{8 \cdot (15 \sqrt{15})}{5^{3}}) + 16 (- \frac{2 \sqrt{15}}{5})$

Step 3.5.2.1.8.6

Multiply $8$ by $15$ .

$f (- \frac{2 \sqrt{15}}{5}) = - \frac{288 \sqrt{15}}{125} - 8 (- \frac{120 \sqrt{15}}{5^{3}}) + 16 (- \frac{2 \sqrt{15}}{5})$

Step 3.5.2.1.9

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

$f (- \frac{2 \sqrt{15}}{5}) = - \frac{288 \sqrt{15}}{125} - 8 (- \frac{120 \sqrt{15}}{125}) + 16 (- \frac{2 \sqrt{15}}{5})$

Step 3.5.2.1.10

Cancel the common factor of $120$ and $125$ .

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Step 3.5.2.1.10.1

Factor $5$ out of $120 \sqrt{15}$ .

$f (- \frac{2 \sqrt{15}}{5}) = - \frac{288 \sqrt{15}}{125} - 8 (- \frac{5 (24 \sqrt{15})}{125}) + 16 (- \frac{2 \sqrt{15}}{5})$

Step 3.5.2.1.10.2

Cancel the common factors.

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Step 3.5.2.1.10.2.1

Factor $5$ out of $125$ .

$f (- \frac{2 \sqrt{15}}{5}) = - \frac{288 \sqrt{15}}{125} - 8 (- \frac{5 (24 \sqrt{15})}{5 (25)}) + 16 (- \frac{2 \sqrt{15}}{5})$

Step 3.5.2.1.10.2.2

Cancel the common factor.

$f (- \frac{2 \sqrt{15}}{5}) = - \frac{288 \sqrt{15}}{125} - 8 (- \frac{5 (24 \sqrt{15})}{5 \cdot 25}) + 16 (- \frac{2 \sqrt{15}}{5})$

Step 3.5.2.1.10.2.3

Rewrite the expression.

$f (- \frac{2 \sqrt{15}}{5}) = - \frac{288 \sqrt{15}}{125} - 8 (- \frac{24 \sqrt{15}}{25}) + 16 (- \frac{2 \sqrt{15}}{5})$

Step 3.5.2.1.11

Multiply $- 8 (- \frac{24 \sqrt{15}}{25})$ .

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Step 3.5.2.1.11.1

Multiply $- 1$ by $- 8$ .

$f (- \frac{2 \sqrt{15}}{5}) = - \frac{288 \sqrt{15}}{125} + 8 (\frac{24 \sqrt{15}}{25}) + 16 (- \frac{2 \sqrt{15}}{5})$

Step 3.5.2.1.11.2

Combine $8$ and $\frac{24 \sqrt{15}}{25}$ .

$f (- \frac{2 \sqrt{15}}{5}) = - \frac{288 \sqrt{15}}{125} + \frac{8 (24 \sqrt{15})}{25} + 16 (- \frac{2 \sqrt{15}}{5})$

Step 3.5.2.1.11.3

Multiply $24$ by $8$ .

$f (- \frac{2 \sqrt{15}}{5}) = - \frac{288 \sqrt{15}}{125} + \frac{192 \sqrt{15}}{25} + 16 (- \frac{2 \sqrt{15}}{5})$

Step 3.5.2.1.12

Multiply $16 (- \frac{2 \sqrt{15}}{5})$ .

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Step 3.5.2.1.12.1

Multiply $- 1$ by $16$ .

$f (- \frac{2 \sqrt{15}}{5}) = - \frac{288 \sqrt{15}}{125} + \frac{192 \sqrt{15}}{25} - 16 \frac{2 \sqrt{15}}{5}$

Step 3.5.2.1.12.2

Combine $- 16$ and $\frac{2 \sqrt{15}}{5}$ .

$f (- \frac{2 \sqrt{15}}{5}) = - \frac{288 \sqrt{15}}{125} + \frac{192 \sqrt{15}}{25} + \frac{- 16 (2 \sqrt{15})}{5}$

Step 3.5.2.1.12.3

Multiply $2$ by $- 16$ .

$f (- \frac{2 \sqrt{15}}{5}) = - \frac{288 \sqrt{15}}{125} + \frac{192 \sqrt{15}}{25} + \frac{- 32 \sqrt{15}}{5}$

Step 3.5.2.1.13

Move the negative in front of the fraction.

$f (- \frac{2 \sqrt{15}}{5}) = - \frac{288 \sqrt{15}}{125} + \frac{192 \sqrt{15}}{25} - \frac{32 \sqrt{15}}{5}$

Step 3.5.2.2

Find the common denominator.

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Step 3.5.2.2.1

Multiply $\frac{192 \sqrt{15}}{25}$ by $\frac{5}{5}$ .

$f (- \frac{2 \sqrt{15}}{5}) = - \frac{288 \sqrt{15}}{125} + \frac{192 \sqrt{15}}{25} \cdot \frac{5}{5} - \frac{32 \sqrt{15}}{5}$

Step 3.5.2.2.2

Multiply $\frac{192 \sqrt{15}}{25}$ by $\frac{5}{5}$ .

$f (- \frac{2 \sqrt{15}}{5}) = - \frac{288 \sqrt{15}}{125} + \frac{192 \sqrt{15} \cdot 5}{25 \cdot 5} - \frac{32 \sqrt{15}}{5}$

Step 3.5.2.2.3

Multiply $\frac{32 \sqrt{15}}{5}$ by $\frac{25}{25}$ .

$f (- \frac{2 \sqrt{15}}{5}) = - \frac{288 \sqrt{15}}{125} + \frac{192 \sqrt{15} \cdot 5}{25 \cdot 5} - (\frac{32 \sqrt{15}}{5} \cdot \frac{25}{25})$

Step 3.5.2.2.4

Multiply $\frac{32 \sqrt{15}}{5}$ by $\frac{25}{25}$ .

$f (- \frac{2 \sqrt{15}}{5}) = - \frac{288 \sqrt{15}}{125} + \frac{192 \sqrt{15} \cdot 5}{25 \cdot 5} - \frac{32 \sqrt{15} \cdot 25}{5 \cdot 25}$

Step 3.5.2.2.5

Reorder the factors of $25 \cdot 5$ .

$f (- \frac{2 \sqrt{15}}{5}) = - \frac{288 \sqrt{15}}{125} + \frac{192 \sqrt{15} \cdot 5}{5 \cdot 25} - \frac{32 \sqrt{15} \cdot 25}{5 \cdot 25}$

Step 3.5.2.2.6

Multiply $5$ by $25$ .

$f (- \frac{2 \sqrt{15}}{5}) = - \frac{288 \sqrt{15}}{125} + \frac{192 \sqrt{15} \cdot 5}{125} - \frac{32 \sqrt{15} \cdot 25}{5 \cdot 25}$

Step 3.5.2.2.7

Multiply $5$ by $25$ .

$f (- \frac{2 \sqrt{15}}{5}) = - \frac{288 \sqrt{15}}{125} + \frac{192 \sqrt{15} \cdot 5}{125} - \frac{32 \sqrt{15} \cdot 25}{125}$

Step 3.5.2.3

Combine the numerators over the common denominator.

$f (- \frac{2 \sqrt{15}}{5}) = \frac{- 288 \sqrt{15} + 192 \sqrt{15} \cdot 5 - 32 \sqrt{15} \cdot 25}{125}$

Step 3.5.2.4

Simplify each term.

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Step 3.5.2.4.1

Multiply $5$ by $192$ .

$f (- \frac{2 \sqrt{15}}{5}) = \frac{- 288 \sqrt{15} + 960 \sqrt{15} - 32 \sqrt{15} \cdot 25}{125}$

Step 3.5.2.4.2

Multiply $25$ by $- 32$ .

$f (- \frac{2 \sqrt{15}}{5}) = \frac{- 288 \sqrt{15} + 960 \sqrt{15} - 800 \sqrt{15}}{125}$

Step 3.5.2.5

Simplify by adding terms.

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Step 3.5.2.5.1

Add $- 288 \sqrt{15}$ and $960 \sqrt{15}$ .

$f (- \frac{2 \sqrt{15}}{5}) = \frac{672 \sqrt{15} - 800 \sqrt{15}}{125}$

Step 3.5.2.5.2

Subtract $800 \sqrt{15}$ from $672 \sqrt{15}$ .

$f (- \frac{2 \sqrt{15}}{5}) = \frac{- 128 \sqrt{15}}{125}$

Step 3.5.2.5.3

Move the negative in front of the fraction.

$f (- \frac{2 \sqrt{15}}{5}) = - \frac{128 \sqrt{15}}{125}$

Step 3.5.2.6

The final answer is $- \frac{128 \sqrt{15}}{125}$ .

$- \frac{128 \sqrt{15}}{125}$

Step 3.6

The point found by substituting $- \frac{2 \sqrt{15}}{5}$ in $f (x) = x^{5} - 8 x^{3} + 16 x$ is $(- \frac{2 \sqrt{15}}{5}, - \frac{128 \sqrt{15}}{125})$ . This point can be an inflection point.

$(- \frac{2 \sqrt{15}}{5}, - \frac{128 \sqrt{15}}{125})$

Step 3.7

Determine the points that could be inflection points.

$(0, 0), (\frac{2 \sqrt{15}}{5}, \frac{128 \sqrt{15}}{125}), (- \frac{2 \sqrt{15}}{5}, - \frac{128 \sqrt{15}}{125})$

Step 4

Split $(- \infty, \infty)$ into intervals around the points that could potentially be inflection points.

$(- \infty, - \frac{2 \sqrt{15}}{5}) \cup (- \frac{2 \sqrt{15}}{5}, 0) \cup (0, \frac{2 \sqrt{15}}{5}) \cup (\frac{2 \sqrt{15}}{5}, \infty)$

Step 5

Substitute a value from the interval $(- \infty, - \frac{2 \sqrt{15}}{5})$ into the second derivative to determine if it is increasing or decreasing.

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Step 5.1

Replace the variable $x$ with $- 1.64919333$ in the expression.

$f'' (- 1.64919333) = 20 {(- 1.64919333)}^{3} - 48 \cdot - 1.64919333$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Raise $- 1.64919333$ to the power of $3$ .

$f'' (- 1.64919333) = 20 \cdot - 4.48553981 - 48 \cdot - 1.64919333$

Step 5.2.1.2

Multiply $20$ by $- 4.48553981$ .

$f'' (- 1.64919333) = - 89.71079625 - 48 \cdot - 1.64919333$

Step 5.2.1.3

Multiply $- 48$ by $- 1.64919333$ .

$f'' (- 1.64919333) = - 89.71079625 + 79.16128024$

Step 5.2.2

Add $- 89.71079625$ and $79.16128024$ .

$f'' (- 1.64919333) = - 10.549516$

Step 5.2.3

The final answer is $- 10.549516$ .

$- 10.549516$

Step 5.3

At $- 1.64919333$ , the second derivative is $- 10.549516$ . Since this is negative, the second derivative is decreasing on the interval $(- \infty, - \frac{2 \sqrt{15}}{5})$

Decreasing on $(- \infty, - \frac{2 \sqrt{15}}{5})$ since $f'' (x) < 0$

Step 6

Substitute a value from the interval $(- \frac{2 \sqrt{15}}{5}, 0)$ into the second derivative to determine if it is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $- 0.77459666$ in the expression.

$f'' (- 0.77459666) = 20 {(- 0.77459666)}^{3} - 48 \cdot - 0.77459666$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Raise $- 0.77459666$ to the power of $3$ .

$f'' (- 0.77459666) = 20 \cdot - 0.464758 - 48 \cdot - 0.77459666$

Step 6.2.1.2

Multiply $20$ by $- 0.464758$ .

$f'' (- 0.77459666) = - 9.29516003 - 48 \cdot - 0.77459666$

Step 6.2.1.3

Multiply $- 48$ by $- 0.77459666$ .

$f'' (- 0.77459666) = - 9.29516003 + 37.18064012$

Step 6.2.2

Add $- 9.29516003$ and $37.18064012$ .

$f'' (- 0.77459666) = 27.88548009$

Step 6.2.3

The final answer is $27.88548009$ .

$27.88548009$

Step 6.3

At $- 0.77459666$ , the second derivative is $27.88548009$ . Since this is positive, the second derivative is increasing on the interval $(- \frac{2 \sqrt{15}}{5}, 0)$ .

Increasing on $(- \frac{2 \sqrt{15}}{5}, 0)$ since $f'' (x) > 0$

Step 7

Substitute a value from the interval $(0, \frac{2 \sqrt{15}}{5})$ into the second derivative to determine if it is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $0.77459666$ in the expression.

$f'' (0.77459666) = 20 {(0.77459666)}^{3} - 48 \cdot 0.77459666$

Step 7.2

Simplify the result.

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Step 7.2.1

Simplify each term.

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Step 7.2.1.1

Raise $0.77459666$ to the power of $3$ .

$f'' (0.77459666) = 20 \cdot 0.464758 - 48 \cdot 0.77459666$

Step 7.2.1.2

Multiply $20$ by $0.464758$ .

$f'' (0.77459666) = 9.29516003 - 48 \cdot 0.77459666$

Step 7.2.1.3

Multiply $- 48$ by $0.77459666$ .

$f'' (0.77459666) = 9.29516003 - 37.18064012$

Step 7.2.2

Subtract $37.18064012$ from $9.29516003$ .

$f'' (0.77459666) = - 27.88548009$

Step 7.2.3

The final answer is $- 27.88548009$ .

$- 27.88548009$

Step 7.3

At $0.77459666$ , the second derivative is $- 27.88548009$ . Since this is negative, the second derivative is decreasing on the interval $(0, \frac{2 \sqrt{15}}{5})$

Decreasing on $(0, \frac{2 \sqrt{15}}{5})$ since $f'' (x) < 0$

Step 8

Substitute a value from the interval $(\frac{2 \sqrt{15}}{5}, \infty)$ into the second derivative to determine if it is increasing or decreasing.

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Step 8.1

Replace the variable $x$ with $1.64919333$ in the expression.

$f'' (1.64919333) = 20 {(1.64919333)}^{3} - 48 \cdot 1.64919333$

Step 8.2

Simplify the result.

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Step 8.2.1

Simplify each term.

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Step 8.2.1.1

Raise $1.64919333$ to the power of $3$ .

$f'' (1.64919333) = 20 \cdot 4.48553981 - 48 \cdot 1.64919333$

Step 8.2.1.2

Multiply $20$ by $4.48553981$ .

$f'' (1.64919333) = 89.71079625 - 48 \cdot 1.64919333$

Step 8.2.1.3

Multiply $- 48$ by $1.64919333$ .

$f'' (1.64919333) = 89.71079625 - 79.16128024$

Step 8.2.2

Subtract $79.16128024$ from $89.71079625$ .

$f'' (1.64919333) = 10.549516$

Step 8.2.3

The final answer is $10.549516$ .

$10.549516$

Step 8.3

At $1.64919333$ , the second derivative is $10.549516$ . Since this is positive, the second derivative is increasing on the interval $(\frac{2 \sqrt{15}}{5}, \infty)$ .

Increasing on $(\frac{2 \sqrt{15}}{5}, \infty)$ since $f'' (x) > 0$

Step 9

An inflection point is a point on a curve at which the concavity changes sign from plus to minus or from minus to plus. The inflection points in this case are $(- \frac{2 \sqrt{15}}{5}, - \frac{128 \sqrt{15}}{125}), (0, 0), (\frac{2 \sqrt{15}}{5}, \frac{128 \sqrt{15}}{125})$ .

$(- \frac{2 \sqrt{15}}{5}, - \frac{128 \sqrt{15}}{125}), (0, 0), (\frac{2 \sqrt{15}}{5}, \frac{128 \sqrt{15}}{125})$

Step 10