Calculus Examples

$f (x) = \frac{1}{15} x^{6} - x^{4}$

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

By the Sum Rule, the derivative of $\frac{1}{15} x^{6} - x^{4}$ with respect to $x$ is $\frac{d}{d x} [\frac{1}{15} x^{6}] + \frac{d}{d x} [- x^{4}]$ .

$\frac{d}{d x} [\frac{1}{15} x^{6}] + \frac{d}{d x} [- x^{4}]$

Step 1.1.2

Evaluate $\frac{d}{d x} [\frac{1}{15} x^{6}]$ .

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Step 1.1.2.1

Since $\frac{1}{15}$ is constant with respect to $x$ , the derivative of $\frac{1}{15} x^{6}$ with respect to $x$ is $\frac{1}{15} \frac{d}{d x} [x^{6}]$ .

$\frac{1}{15} \frac{d}{d x} [x^{6}] + \frac{d}{d x} [- x^{4}]$

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 $n = 6$ .

$\frac{1}{15} (6 x^{5}) + \frac{d}{d x} [- x^{4}]$

Step 1.1.2.3

Combine $6$ and $\frac{1}{15}$ .

$\frac{6}{15} x^{5} + \frac{d}{d x} [- x^{4}]$

Step 1.1.2.4

Combine $\frac{6}{15}$ and $x^{5}$ .

$\frac{6 x^{5}}{15} + \frac{d}{d x} [- x^{4}]$

Step 1.1.2.5

Cancel the common factor of $6$ and $15$ .

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Step 1.1.2.5.1

Factor $3$ out of $6 x^{5}$ .

$\frac{3 (2 x^{5})}{15} + \frac{d}{d x} [- x^{4}]$

Step 1.1.2.5.2

Cancel the common factors.

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Step 1.1.2.5.2.1

Factor $3$ out of $15$ .

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

Step 1.1.2.5.2.2

Cancel the common factor.

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

Step 1.1.2.5.2.3

Rewrite the expression.

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

Step 1.1.3

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

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Step 1.1.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{4}$ with respect to $x$ is $- \frac{d}{d x} [x^{4}]$ .

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

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 $n = 4$ .

$\frac{2 x^{5}}{5} - (4 x^{3})$

Step 1.1.3.3

Multiply $4$ by $- 1$ .

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

Step 1.2

Find the second derivative.

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Step 1.2.1

By the Sum Rule, the derivative of $\frac{2 x^{5}}{5} - 4 x^{3}$ with respect to $x$ is $\frac{d}{d x} [\frac{2 x^{5}}{5}] + \frac{d}{d x} [- 4 x^{3}]$ .

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

Step 1.2.2

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

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Step 1.2.2.1

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

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

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 $n = 5$ .

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

Step 1.2.2.3

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

$\frac{5 \cdot 2}{5} x^{4} + \frac{d}{d x} [- 4 x^{3}]$

Step 1.2.2.4

Multiply $5$ by $2$ .

$\frac{10}{5} x^{4} + \frac{d}{d x} [- 4 x^{3}]$

Step 1.2.2.5

Combine $\frac{10}{5}$ and $x^{4}$ .

$\frac{10 x^{4}}{5} + \frac{d}{d x} [- 4 x^{3}]$

Step 1.2.2.6

Cancel the common factor of $10$ and $5$ .

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Step 1.2.2.6.1

Factor $5$ out of $10 x^{4}$ .

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

Step 1.2.2.6.2

Cancel the common factors.

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Step 1.2.2.6.2.1

Factor $5$ out of $5$ .

$\frac{5 (2 x^{4})}{5 (1)} + \frac{d}{d x} [- 4 x^{3}]$

Step 1.2.2.6.2.2

Cancel the common factor.

$\frac{5 (2 x^{4})}{5 \cdot 1} + \frac{d}{d x} [- 4 x^{3}]$

Step 1.2.2.6.2.3

Rewrite the expression.

$\frac{2 x^{4}}{1} + \frac{d}{d x} [- 4 x^{3}]$

Step 1.2.2.6.2.4

Divide $2 x^{4}$ by $1$ .

$2 x^{4} + \frac{d}{d x} [- 4 x^{3}]$

Step 1.2.3

Evaluate $\frac{d}{d x} [- 4 x^{3}]$ .

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Step 1.2.3.1

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 x^{3}$ with respect to $x$ is $- 4 \frac{d}{d x} [x^{3}]$ .

$2 x^{4} - 4 \frac{d}{d x} [x^{3}]$

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 $n = 3$ .

$2 x^{4} - 4 (3 x^{2})$

Step 1.2.3.3

Multiply $3$ by $- 4$ .

$f'' (x) = 2 x^{4} - 12 x^{2}$

Step 1.3

The second derivative of $f (x)$ with respect to $x$ is $2 x^{4} - 12 x^{2}$ .

$2 x^{4} - 12 x^{2}$

Step 2

Set the second derivative equal to $0$ then solve the equation $2 x^{4} - 12 x^{2} = 0$ .

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Step 2.1

Set the second derivative equal to $0$ .

$2 x^{4} - 12 x^{2} = 0$

Step 2.2

Factor the left side of the equation.

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Step 2.2.1

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

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

Step 2.2.2

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$2 u^{2} - 12 u = 0$

Step 2.2.3

Factor $2 u$ out of $2 u^{2} - 12 u$ .

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Step 2.2.3.1

Factor $2 u$ out of $2 u^{2}$ .

$2 u (u) - 12 u = 0$

Step 2.2.3.2

Factor $2 u$ out of $- 12 u$ .

$2 u (u) + 2 u (- 6) = 0$

Step 2.2.3.3

Factor $2 u$ out of $2 u (u) + 2 u (- 6)$ .

$2 u (u - 6) = 0$

Step 2.2.4

Replace all occurrences of $u$ with $x^{2}$ .

$2 x^{2} (x^{2} - 6) = 0$

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 $0$ .

$x^{2} = 0$

$x^{2} - 6 = 0$

Step 2.4

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

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Step 2.4.1

Set $x^{2}$ equal to $0$ .

$x^{2} = 0$

Step 2.4.2

Solve $x^{2} = 0$ for $x$ .

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Step 2.4.2.1

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

$x = \pm \sqrt{0}$

Step 2.4.2.2

Simplify $\pm \sqrt{0}$ .

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Step 2.4.2.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 2.4.2.2.2

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

$x = \pm 0$

Step 2.4.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 2.5

Set $x^{2} - 6$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $x^{2} - 6$ equal to $0$ .

$x^{2} - 6 = 0$

Step 2.5.2

Solve $x^{2} - 6 = 0$ for $x$ .

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Step 2.5.2.1

Add $6$ to both sides of the equation.

$x^{2} = 6$

Step 2.5.2.2

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

$x = \pm \sqrt{6}$

Step 2.5.2.3

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

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Step 2.5.2.3.1

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

$x = \sqrt{6}$

Step 2.5.2.3.2

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

$x = - \sqrt{6}$

Step 2.5.2.3.3

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

$x = \sqrt{6}, - \sqrt{6}$

Step 2.6

The final solution is all the values that make $2 x^{2} (x^{2} - 6) = 0$ true.

$x = 0, \sqrt{6}, - \sqrt{6}$

Step 3

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

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Step 3.1

Substitute $0$ in $f (x) = \frac{1}{15} x^{6} - x^{4}$ to find the value of $y$ .

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Step 3.1.1

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

$f (0) = \frac{1}{15} \cdot {(0)}^{6} - {(0)}^{4}$

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 $0$ .

$f (0) = \frac{1}{15} \cdot 0 - {(0)}^{4}$

Step 3.1.2.1.2

Multiply $\frac{1}{15}$ by $0$ .

$f (0) = 0 - {(0)}^{4}$

Step 3.1.2.1.3

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

$f (0) = 0 - 0$

Step 3.1.2.1.4

Multiply $- 1$ by $0$ .

$f (0) = 0 + 0$

Step 3.1.2.2

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 $f (x) = \frac{1}{15} x^{6} - x^{4}$ is $(0, 0)$ . This point can be an inflection point.

$(0, 0)$

Step 3.3

Substitute $\sqrt{6}$ in $f (x) = \frac{1}{15} x^{6} - x^{4}$ to find the value of $y$ .

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Step 3.3.1

Replace the variable $x$ with $\sqrt{6}$ in the expression.

$f (\sqrt{6}) = \frac{1}{15} \cdot {(\sqrt{6})}^{6} - {(\sqrt{6})}^{4}$

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

Rewrite ${\sqrt{6}}^{6}$ as $6^{3}$ .

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Step 3.3.2.1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$f (\sqrt{6}) = \frac{1}{15} \cdot {(6^{\frac{1}{2}})}^{6} - {(\sqrt{6})}^{4}$

Step 3.3.2.1.1.2

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

$f (\sqrt{6}) = \frac{1}{15} \cdot 6^{\frac{1}{2} \cdot 6} - {(\sqrt{6})}^{4}$

Step 3.3.2.1.1.3

Combine $\frac{1}{2}$ and $6$ .

$f (\sqrt{6}) = \frac{1}{15} \cdot 6^{\frac{6}{2}} - {(\sqrt{6})}^{4}$

Step 3.3.2.1.1.4

Cancel the common factor of $6$ and $2$ .

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Step 3.3.2.1.1.4.1

Factor $2$ out of $6$ .

$f (\sqrt{6}) = \frac{1}{15} \cdot 6^{\frac{2 \cdot 3}{2}} - {(\sqrt{6})}^{4}$

Step 3.3.2.1.1.4.2

Cancel the common factors.

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Step 3.3.2.1.1.4.2.1

Factor $2$ out of $2$ .

$f (\sqrt{6}) = \frac{1}{15} \cdot 6^{\frac{2 \cdot 3}{2 (1)}} - {(\sqrt{6})}^{4}$

Step 3.3.2.1.1.4.2.2

Cancel the common factor.

$f (\sqrt{6}) = \frac{1}{15} \cdot 6^{\frac{2 \cdot 3}{2 \cdot 1}} - {(\sqrt{6})}^{4}$

Step 3.3.2.1.1.4.2.3

Rewrite the expression.

$f (\sqrt{6}) = \frac{1}{15} \cdot 6^{\frac{3}{1}} - {(\sqrt{6})}^{4}$

Step 3.3.2.1.1.4.2.4

Divide $3$ by $1$ .

$f (\sqrt{6}) = \frac{1}{15} \cdot 6^{3} - {(\sqrt{6})}^{4}$

Step 3.3.2.1.2

Raise $6$ to the power of $3$ .

$f (\sqrt{6}) = \frac{1}{15} \cdot 216 - {(\sqrt{6})}^{4}$

Step 3.3.2.1.3

Cancel the common factor of $3$ .

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Step 3.3.2.1.3.1

Factor $3$ out of $15$ .

$f (\sqrt{6}) = \frac{1}{3 (5)} \cdot 216 - {(\sqrt{6})}^{4}$

Step 3.3.2.1.3.2

Factor $3$ out of $216$ .

$f (\sqrt{6}) = \frac{1}{3 \cdot 5} \cdot (3 \cdot 72) - {(\sqrt{6})}^{4}$

Step 3.3.2.1.3.3

Cancel the common factor.

$f (\sqrt{6}) = \frac{1}{3 \cdot 5} \cdot (3 \cdot 72) - {(\sqrt{6})}^{4}$

Step 3.3.2.1.3.4

Rewrite the expression.

$f (\sqrt{6}) = \frac{1}{5} \cdot 72 - {(\sqrt{6})}^{4}$

Step 3.3.2.1.4

Combine $\frac{1}{5}$ and $72$ .

$f (\sqrt{6}) = \frac{72}{5} - {(\sqrt{6})}^{4}$

Step 3.3.2.1.5

Rewrite ${\sqrt{6}}^{4}$ as $6^{2}$ .

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Step 3.3.2.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$f (\sqrt{6}) = \frac{72}{5} - {(6^{\frac{1}{2}})}^{4}$

Step 3.3.2.1.5.2

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

$f (\sqrt{6}) = \frac{72}{5} - 6^{\frac{1}{2} \cdot 4}$

Step 3.3.2.1.5.3

Combine $\frac{1}{2}$ and $4$ .

$f (\sqrt{6}) = \frac{72}{5} - 6^{\frac{4}{2}}$

Step 3.3.2.1.5.4

Cancel the common factor of $4$ and $2$ .

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Step 3.3.2.1.5.4.1

Factor $2$ out of $4$ .

$f (\sqrt{6}) = \frac{72}{5} - 6^{\frac{2 \cdot 2}{2}}$

Step 3.3.2.1.5.4.2

Cancel the common factors.

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Step 3.3.2.1.5.4.2.1

Factor $2$ out of $2$ .

$f (\sqrt{6}) = \frac{72}{5} - 6^{\frac{2 \cdot 2}{2 (1)}}$

Step 3.3.2.1.5.4.2.2

Cancel the common factor.

$f (\sqrt{6}) = \frac{72}{5} - 6^{\frac{2 \cdot 2}{2 \cdot 1}}$

Step 3.3.2.1.5.4.2.3

Rewrite the expression.

$f (\sqrt{6}) = \frac{72}{5} - 6^{\frac{2}{1}}$

Step 3.3.2.1.5.4.2.4

Divide $2$ by $1$ .

$f (\sqrt{6}) = \frac{72}{5} - 6^{2}$

Step 3.3.2.1.6

Raise $6$ to the power of $2$ .

$f (\sqrt{6}) = \frac{72}{5} - 1 \cdot 36$

Step 3.3.2.1.7

Multiply $- 1$ by $36$ .

$f (\sqrt{6}) = \frac{72}{5} - 36$

Step 3.3.2.2

To write $- 36$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$f (\sqrt{6}) = \frac{72}{5} - 36 \cdot \frac{5}{5}$

Step 3.3.2.3

Combine $- 36$ and $\frac{5}{5}$ .

$f (\sqrt{6}) = \frac{72}{5} + \frac{- 36 \cdot 5}{5}$

Step 3.3.2.4

Combine the numerators over the common denominator.

$f (\sqrt{6}) = \frac{72 - 36 \cdot 5}{5}$

Step 3.3.2.5

Simplify the numerator.

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Step 3.3.2.5.1

Multiply $- 36$ by $5$ .

$f (\sqrt{6}) = \frac{72 - 180}{5}$

Step 3.3.2.5.2

Subtract $180$ from $72$ .

$f (\sqrt{6}) = \frac{- 108}{5}$

Step 3.3.2.6

Move the negative in front of the fraction.

$f (\sqrt{6}) = - \frac{108}{5}$

Step 3.3.2.7

The final answer is $- \frac{108}{5}$ .

$- \frac{108}{5}$

Step 3.4

The point found by substituting $\sqrt{6}$ in $f (x) = \frac{1}{15} x^{6} - x^{4}$ is $(\sqrt{6}, - \frac{108}{5})$ . This point can be an inflection point.

$(\sqrt{6}, - \frac{108}{5})$

Step 3.5

Substitute $- \sqrt{6}$ in $f (x) = \frac{1}{15} x^{6} - x^{4}$ to find the value of $y$ .

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Step 3.5.1

Replace the variable $x$ with $- \sqrt{6}$ in the expression.

$f (- \sqrt{6}) = \frac{1}{15} \cdot {(- \sqrt{6})}^{6} - {(- \sqrt{6})}^{4}$

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

Apply the product rule to $- \sqrt{6}$ .

$f (- \sqrt{6}) = \frac{1}{15} \cdot ({(- 1)}^{6} {\sqrt{6}}^{6}) - {(- \sqrt{6})}^{4}$

Step 3.5.2.1.2

Raise $- 1$ to the power of $6$ .

$f (- \sqrt{6}) = \frac{1}{15} \cdot (1 {\sqrt{6}}^{6}) - {(- \sqrt{6})}^{4}$

Step 3.5.2.1.3

Multiply ${\sqrt{6}}^{6}$ by $1$ .

$f (- \sqrt{6}) = \frac{1}{15} \cdot {\sqrt{6}}^{6} - {(- \sqrt{6})}^{4}$

Step 3.5.2.1.4

Rewrite ${\sqrt{6}}^{6}$ as $6^{3}$ .

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Step 3.5.2.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$f (- \sqrt{6}) = \frac{1}{15} \cdot {(6^{\frac{1}{2}})}^{6} - {(- \sqrt{6})}^{4}$

Step 3.5.2.1.4.2

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

$f (- \sqrt{6}) = \frac{1}{15} \cdot 6^{\frac{1}{2} \cdot 6} - {(- \sqrt{6})}^{4}$

Step 3.5.2.1.4.3

Combine $\frac{1}{2}$ and $6$ .

$f (- \sqrt{6}) = \frac{1}{15} \cdot 6^{\frac{6}{2}} - {(- \sqrt{6})}^{4}$

Step 3.5.2.1.4.4

Cancel the common factor of $6$ and $2$ .

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Step 3.5.2.1.4.4.1

Factor $2$ out of $6$ .

$f (- \sqrt{6}) = \frac{1}{15} \cdot 6^{\frac{2 \cdot 3}{2}} - {(- \sqrt{6})}^{4}$

Step 3.5.2.1.4.4.2

Cancel the common factors.

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Step 3.5.2.1.4.4.2.1

Factor $2$ out of $2$ .

$f (- \sqrt{6}) = \frac{1}{15} \cdot 6^{\frac{2 \cdot 3}{2 (1)}} - {(- \sqrt{6})}^{4}$

Step 3.5.2.1.4.4.2.2

Cancel the common factor.

$f (- \sqrt{6}) = \frac{1}{15} \cdot 6^{\frac{2 \cdot 3}{2 \cdot 1}} - {(- \sqrt{6})}^{4}$

Step 3.5.2.1.4.4.2.3

Rewrite the expression.

$f (- \sqrt{6}) = \frac{1}{15} \cdot 6^{\frac{3}{1}} - {(- \sqrt{6})}^{4}$

Step 3.5.2.1.4.4.2.4

Divide $3$ by $1$ .

$f (- \sqrt{6}) = \frac{1}{15} \cdot 6^{3} - {(- \sqrt{6})}^{4}$

Step 3.5.2.1.5

Raise $6$ to the power of $3$ .

$f (- \sqrt{6}) = \frac{1}{15} \cdot 216 - {(- \sqrt{6})}^{4}$

Step 3.5.2.1.6

Cancel the common factor of $3$ .

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Step 3.5.2.1.6.1

Factor $3$ out of $15$ .

$f (- \sqrt{6}) = \frac{1}{3 (5)} \cdot 216 - {(- \sqrt{6})}^{4}$

Step 3.5.2.1.6.2

Factor $3$ out of $216$ .

$f (- \sqrt{6}) = \frac{1}{3 \cdot 5} \cdot (3 \cdot 72) - {(- \sqrt{6})}^{4}$

Step 3.5.2.1.6.3

Cancel the common factor.

$f (- \sqrt{6}) = \frac{1}{3 \cdot 5} \cdot (3 \cdot 72) - {(- \sqrt{6})}^{4}$

Step 3.5.2.1.6.4

Rewrite the expression.

$f (- \sqrt{6}) = \frac{1}{5} \cdot 72 - {(- \sqrt{6})}^{4}$

Step 3.5.2.1.7

Combine $\frac{1}{5}$ and $72$ .

$f (- \sqrt{6}) = \frac{72}{5} - {(- \sqrt{6})}^{4}$

Step 3.5.2.1.8

Apply the product rule to $- \sqrt{6}$ .

$f (- \sqrt{6}) = \frac{72}{5} - ({(- 1)}^{4} {\sqrt{6}}^{4})$

Step 3.5.2.1.9

Multiply $- 1$ by ${(- 1)}^{4}$ by adding the exponents.

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Step 3.5.2.1.9.1

Move ${(- 1)}^{4}$ .

$f (- \sqrt{6}) = \frac{72}{5} + {(- 1)}^{4} \cdot (- 1 {\sqrt{6}}^{4})$

Step 3.5.2.1.9.2

Multiply ${(- 1)}^{4}$ by $- 1$ .

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Step 3.5.2.1.9.2.1

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

$f (- \sqrt{6}) = \frac{72}{5} + {(- 1)}^{4} \cdot ((- 1) {\sqrt{6}}^{4})$

Step 3.5.2.1.9.2.2

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

$f (- \sqrt{6}) = \frac{72}{5} + {(- 1)}^{4 + 1} {\sqrt{6}}^{4}$

Step 3.5.2.1.9.3

Add $4$ and $1$ .

$f (- \sqrt{6}) = \frac{72}{5} + {(- 1)}^{5} {\sqrt{6}}^{4}$

Step 3.5.2.1.10

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

$f (- \sqrt{6}) = \frac{72}{5} - {\sqrt{6}}^{4}$

Step 3.5.2.1.11

Rewrite ${\sqrt{6}}^{4}$ as $6^{2}$ .

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Step 3.5.2.1.11.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$f (- \sqrt{6}) = \frac{72}{5} - {(6^{\frac{1}{2}})}^{4}$

Step 3.5.2.1.11.2

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

$f (- \sqrt{6}) = \frac{72}{5} - 6^{\frac{1}{2} \cdot 4}$

Step 3.5.2.1.11.3

Combine $\frac{1}{2}$ and $4$ .

$f (- \sqrt{6}) = \frac{72}{5} - 6^{\frac{4}{2}}$

Step 3.5.2.1.11.4

Cancel the common factor of $4$ and $2$ .

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Step 3.5.2.1.11.4.1

Factor $2$ out of $4$ .

$f (- \sqrt{6}) = \frac{72}{5} - 6^{\frac{2 \cdot 2}{2}}$

Step 3.5.2.1.11.4.2

Cancel the common factors.

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Step 3.5.2.1.11.4.2.1

Factor $2$ out of $2$ .

$f (- \sqrt{6}) = \frac{72}{5} - 6^{\frac{2 \cdot 2}{2 (1)}}$

Step 3.5.2.1.11.4.2.2

Cancel the common factor.

$f (- \sqrt{6}) = \frac{72}{5} - 6^{\frac{2 \cdot 2}{2 \cdot 1}}$

Step 3.5.2.1.11.4.2.3

Rewrite the expression.

$f (- \sqrt{6}) = \frac{72}{5} - 6^{\frac{2}{1}}$

Step 3.5.2.1.11.4.2.4

Divide $2$ by $1$ .

$f (- \sqrt{6}) = \frac{72}{5} - 6^{2}$

Step 3.5.2.1.12

Raise $6$ to the power of $2$ .

$f (- \sqrt{6}) = \frac{72}{5} - 1 \cdot 36$

Step 3.5.2.1.13

Multiply $- 1$ by $36$ .

$f (- \sqrt{6}) = \frac{72}{5} - 36$

Step 3.5.2.2

To write $- 36$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$f (- \sqrt{6}) = \frac{72}{5} - 36 \cdot \frac{5}{5}$

Step 3.5.2.3

Combine $- 36$ and $\frac{5}{5}$ .

$f (- \sqrt{6}) = \frac{72}{5} + \frac{- 36 \cdot 5}{5}$

Step 3.5.2.4

Combine the numerators over the common denominator.

$f (- \sqrt{6}) = \frac{72 - 36 \cdot 5}{5}$

Step 3.5.2.5

Simplify the numerator.

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Step 3.5.2.5.1

Multiply $- 36$ by $5$ .

$f (- \sqrt{6}) = \frac{72 - 180}{5}$

Step 3.5.2.5.2

Subtract $180$ from $72$ .

$f (- \sqrt{6}) = \frac{- 108}{5}$

Step 3.5.2.6

Move the negative in front of the fraction.

$f (- \sqrt{6}) = - \frac{108}{5}$

Step 3.5.2.7

The final answer is $- \frac{108}{5}$ .

$- \frac{108}{5}$

Step 3.6

The point found by substituting $- \sqrt{6}$ in $f (x) = \frac{1}{15} x^{6} - x^{4}$ is $(- \sqrt{6}, - \frac{108}{5})$ . This point can be an inflection point.

$(- \sqrt{6}, - \frac{108}{5})$

Step 3.7

Determine the points that could be inflection points.

$(0, 0), (\sqrt{6}, - \frac{108}{5}), (- \sqrt{6}, - \frac{108}{5})$

Step 4

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

$(- \infty, - \sqrt{6}) \cup (- \sqrt{6}, 0) \cup (0, \sqrt{6}) \cup (\sqrt{6}, \infty)$

Step 5

Substitute a value from the interval $(- \infty, - \sqrt{6})$ into the second derivative to determine if it is increasing or decreasing.

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Step 5.1

Replace the variable $x$ with $- 2.54948974$ in the expression.

$f'' (- 2.54948974) = 2 {(- 2.54948974)}^{4} - 12 {(- 2.54948974)}^{2}$

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 $- 2.54948974$ to the power of $4$ .

$f'' (- 2.54948974) = 2 \cdot 42.24867334 - 12 {(- 2.54948974)}^{2}$

Step 5.2.1.2

Multiply $2$ by $42.24867334$ .

$f'' (- 2.54948974) = 84.49734668 - 12 {(- 2.54948974)}^{2}$

Step 5.2.1.3

Raise $- 2.54948974$ to the power of $2$ .

$f'' (- 2.54948974) = 84.49734668 - 12 \cdot 6.49989794$

Step 5.2.1.4

Multiply $- 12$ by $6.49989794$ .

$f'' (- 2.54948974) = 84.49734668 - 77.99877538$

Step 5.2.2

Subtract $77.99877538$ from $84.49734668$ .

$f'' (- 2.54948974) = 6.4985713$

Step 5.2.3

The final answer is $6.4985713$ .

$6.4985713$

Step 5.3

At $- 2.54948974$ , the second derivative is $6.4985713$ . Since this is positive, the second derivative is increasing on the interval $(- \infty, - \sqrt{6})$ .

Increasing on $(- \infty, - \sqrt{6})$ since $f'' (x) > 0$

Step 6

Substitute a value from the interval $(- \sqrt{6}, 0)$ into the second derivative to determine if it is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $- 1.22474487$ in the expression.

$f'' (- 1.22474487) = 2 {(- 1.22474487)}^{4} - 12 {(- 1.22474487)}^{2}$

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 $- 1.22474487$ to the power of $4$ .

$f'' (- 1.22474487) = 2 \cdot 2.25 - 12 {(- 1.22474487)}^{2}$

Step 6.2.1.2

Multiply $2$ by $2.25$ .

$f'' (- 1.22474487) = 4.5 - 12 {(- 1.22474487)}^{2}$

Step 6.2.1.3

Raise $- 1.22474487$ to the power of $2$ .

$f'' (- 1.22474487) = 4.5 - 12 \cdot 1.5$

Step 6.2.1.4

Multiply $- 12$ by $1.5$ .

$f'' (- 1.22474487) = 4.5 - 18$

Step 6.2.2

Subtract $18$ from $4.5$ .

$f'' (- 1.22474487) = - 13.5$

Step 6.2.3

The final answer is $- 13.5$ .

$- 13.5$

Step 6.3

At $- 1.22474487$ , the second derivative is $- 13.5$ . Since this is negative, the second derivative is decreasing on the interval $(- \sqrt{6}, 0)$

Decreasing on $(- \sqrt{6}, 0)$ since $f'' (x) < 0$

Step 7

Substitute a value from the interval $(0, \sqrt{6})$ into the second derivative to determine if it is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $1.22474487$ in the expression.

$f'' (1.22474487) = 2 {(1.22474487)}^{4} - 12 {(1.22474487)}^{2}$

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 $1.22474487$ to the power of $4$ .

$f'' (1.22474487) = 2 \cdot 2.25 - 12 {(1.22474487)}^{2}$

Step 7.2.1.2

Multiply $2$ by $2.25$ .

$f'' (1.22474487) = 4.5 - 12 {(1.22474487)}^{2}$

Step 7.2.1.3

Raise $1.22474487$ to the power of $2$ .

$f'' (1.22474487) = 4.5 - 12 \cdot 1.5$

Step 7.2.1.4

Multiply $- 12$ by $1.5$ .

$f'' (1.22474487) = 4.5 - 18$

Step 7.2.2

Subtract $18$ from $4.5$ .

$f'' (1.22474487) = - 13.5$

Step 7.2.3

The final answer is $- 13.5$ .

$- 13.5$

Step 7.3

At $1.22474487$ , the second derivative is $- 13.5$ . Since this is negative, the second derivative is decreasing on the interval $(0, \sqrt{6})$

Decreasing on $(0, \sqrt{6})$ since $f'' (x) < 0$

Step 8

Substitute a value from the interval $(\sqrt{6}, \infty)$ into the second derivative to determine if it is increasing or decreasing.

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Step 8.1

Replace the variable $x$ with $2.54948974$ in the expression.

$f'' (2.54948974) = 2 {(2.54948974)}^{4} - 12 {(2.54948974)}^{2}$

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 $2.54948974$ to the power of $4$ .

$f'' (2.54948974) = 2 \cdot 42.24867334 - 12 {(2.54948974)}^{2}$

Step 8.2.1.2

Multiply $2$ by $42.24867334$ .

$f'' (2.54948974) = 84.49734668 - 12 {(2.54948974)}^{2}$

Step 8.2.1.3

Raise $2.54948974$ to the power of $2$ .

$f'' (2.54948974) = 84.49734668 - 12 \cdot 6.49989794$

Step 8.2.1.4

Multiply $- 12$ by $6.49989794$ .

$f'' (2.54948974) = 84.49734668 - 77.99877538$

Step 8.2.2

Subtract $77.99877538$ from $84.49734668$ .

$f'' (2.54948974) = 6.4985713$

Step 8.2.3

The final answer is $6.4985713$ .

$6.4985713$

Step 8.3

At $2.54948974$ , the second derivative is $6.4985713$ . Since this is positive, the second derivative is increasing on the interval $(\sqrt{6}, \infty)$ .

Increasing on $(\sqrt{6}, \infty)$ 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 $(- \sqrt{6}, - \frac{108}{5}), (\sqrt{6}, - \frac{108}{5})$ .

$(- \sqrt{6}, - \frac{108}{5}), (\sqrt{6}, - \frac{108}{5})$

Step 10