Calculus Examples

$f (x) = - \frac{2}{5} x^{6} + 5 x^{4}$

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 $- \frac{2}{5} x^{6} + 5 x^{4}$ with respect to $x$ is $\frac{d}{d x} [- \frac{2}{5} x^{6}] + \frac{d}{d x} [5 x^{4}]$ .

$\frac{d}{d x} [- \frac{2}{5} x^{6}] + \frac{d}{d x} [5 x^{4}]$

Step 1.1.2

Evaluate $\frac{d}{d x} [- \frac{2}{5} x^{6}]$ .

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Step 1.1.2.1

Since $- \frac{2}{5}$ is constant with respect to $x$ , the derivative of $- \frac{2}{5} x^{6}$ with respect to $x$ is $- \frac{2}{5} \frac{d}{d x} [x^{6}]$ .

$- \frac{2}{5} \frac{d}{d x} [x^{6}] + \frac{d}{d x} [5 x^{4}]$

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 $n = 6$ .

$- \frac{2}{5} (6 x^{5}) + \frac{d}{d x} [5 x^{4}]$

Step 1.1.2.3

Multiply $6$ by $- 1$ .

$- 6 (\frac{2}{5}) x^{5} + \frac{d}{d x} [5 x^{4}]$

Step 1.1.2.4

Combine $- 6$ and $\frac{2}{5}$ .

$\frac{- 6 \cdot 2}{5} x^{5} + \frac{d}{d x} [5 x^{4}]$

Step 1.1.2.5

Multiply $- 6$ by $2$ .

$\frac{- 12}{5} x^{5} + \frac{d}{d x} [5 x^{4}]$

Step 1.1.2.6

Combine $\frac{- 12}{5}$ and $x^{5}$ .

$\frac{- 12 x^{5}}{5} + \frac{d}{d x} [5 x^{4}]$

Step 1.1.2.7

Move the negative in front of the fraction.

$- \frac{12 x^{5}}{5} + \frac{d}{d x} [5 x^{4}]$

Step 1.1.3

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

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Step 1.1.3.1

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

$- \frac{12 x^{5}}{5} + 5 \frac{d}{d x} [x^{4}]$

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 $n = 4$ .

$- \frac{12 x^{5}}{5} + 5 (4 x^{3})$

Step 1.1.3.3

Multiply $4$ by $5$ .

$f' (x) = - \frac{12 x^{5}}{5} + 20 x^{3}$

Step 1.2

Find the second derivative.

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Step 1.2.1

By the Sum Rule, the derivative of $- \frac{12 x^{5}}{5} + 20 x^{3}$ with respect to $x$ is $\frac{d}{d x} [- \frac{12 x^{5}}{5}] + \frac{d}{d x} [20 x^{3}]$ .

$\frac{d}{d x} [- \frac{12 x^{5}}{5}] + \frac{d}{d x} [20 x^{3}]$

Step 1.2.2

Evaluate $\frac{d}{d x} [- \frac{12 x^{5}}{5}]$ .

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Step 1.2.2.1

Since $- \frac{12}{5}$ is constant with respect to $x$ , the derivative of $- \frac{12 x^{5}}{5}$ with respect to $x$ is $- \frac{12}{5} \frac{d}{d x} [x^{5}]$ .

$- \frac{12}{5} \frac{d}{d x} [x^{5}] + \frac{d}{d x} [20 x^{3}]$

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 $n = 5$ .

$- \frac{12}{5} (5 x^{4}) + \frac{d}{d x} [20 x^{3}]$

Step 1.2.2.3

Multiply $5$ by $- 1$ .

$- 5 (\frac{12}{5}) x^{4} + \frac{d}{d x} [20 x^{3}]$

Step 1.2.2.4

Combine $- 5$ and $\frac{12}{5}$ .

$\frac{- 5 \cdot 12}{5} x^{4} + \frac{d}{d x} [20 x^{3}]$

Step 1.2.2.5

Multiply $- 5$ by $12$ .

$\frac{- 60}{5} x^{4} + \frac{d}{d x} [20 x^{3}]$

Step 1.2.2.6

Combine $\frac{- 60}{5}$ and $x^{4}$ .

$\frac{- 60 x^{4}}{5} + \frac{d}{d x} [20 x^{3}]$

Step 1.2.2.7

Cancel the common factor of $- 60$ and $5$ .

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Step 1.2.2.7.1

Factor $5$ out of $- 60 x^{4}$ .

$\frac{5 (- 12 x^{4})}{5} + \frac{d}{d x} [20 x^{3}]$

Step 1.2.2.7.2

Cancel the common factors.

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Step 1.2.2.7.2.1

Factor $5$ out of $5$ .

$\frac{5 (- 12 x^{4})}{5 (1)} + \frac{d}{d x} [20 x^{3}]$

Step 1.2.2.7.2.2

Cancel the common factor.

$\frac{5 (- 12 x^{4})}{5 \cdot 1} + \frac{d}{d x} [20 x^{3}]$

Step 1.2.2.7.2.3

Rewrite the expression.

$\frac{- 12 x^{4}}{1} + \frac{d}{d x} [20 x^{3}]$

Step 1.2.2.7.2.4

Divide $- 12 x^{4}$ by $1$ .

$- 12 x^{4} + \frac{d}{d x} [20 x^{3}]$

Step 1.2.3

Evaluate $\frac{d}{d x} [20 x^{3}]$ .

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Step 1.2.3.1

Since $20$ is constant with respect to $x$ , the derivative of $20 x^{3}$ with respect to $x$ is $20 \frac{d}{d x} [x^{3}]$ .

$- 12 x^{4} + 20 \frac{d}{d x} [x^{3}]$

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 $n = 3$ .

$- 12 x^{4} + 20 (3 x^{2})$

Step 1.2.3.3

Multiply $3$ by $20$ .

$f'' (x) = - 12 x^{4} + 60 x^{2}$

Step 1.3

The second derivative of $f (x)$ with respect to $x$ is $- 12 x^{4} + 60 x^{2}$ .

$- 12 x^{4} + 60 x^{2}$

Step 2

Set the second derivative equal to $0$ then solve the equation $- 12 x^{4} + 60 x^{2} = 0$ .

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Step 2.1

Set the second derivative equal to $0$ .

$- 12 x^{4} + 60 x^{2} = 0$

Step 2.2

Factor the left side of the equation.

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Step 2.2.1

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

$- 12 {(x^{2})}^{2} + 60 x^{2} = 0$

Step 2.2.2

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

$- 12 u^{2} + 60 u = 0$

Step 2.2.3

Factor $12 u$ out of $- 12 u^{2} + 60 u$ .

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Step 2.2.3.1

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

$12 u (- u) + 60 u = 0$

Step 2.2.3.2

Factor $12 u$ out of $60 u$ .

$12 u (- u) + 12 u (5) = 0$

Step 2.2.3.3

Factor $12 u$ out of $12 u (- u) + 12 u (5)$ .

$12 u (- u + 5) = 0$

Step 2.2.4

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

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

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 $0$ .

$x^{2} = 0$

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

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

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Step 2.5.1

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

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

Step 2.5.2

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

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Step 2.5.2.1

Subtract $5$ from both sides of the equation.

$- x^{2} = - 5$

Step 2.5.2.2

Divide each term in $- x^{2} = - 5$ by $- 1$ and simplify.

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Step 2.5.2.2.1

Divide each term in $- x^{2} = - 5$ by $- 1$ .

$\frac{- x^{2}}{- 1} = \frac{- 5}{- 1}$

Step 2.5.2.2.2

Simplify the left side.

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Step 2.5.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} = \frac{- 5}{- 1}$

Step 2.5.2.2.2.2

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

$x^{2} = \frac{- 5}{- 1}$

Step 2.5.2.2.3

Simplify the right side.

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Step 2.5.2.2.3.1

Divide $- 5$ by $- 1$ .

$x^{2} = 5$

Step 2.5.2.3

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

$x = \pm \sqrt{5}$

Step 2.5.2.4

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

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Step 2.5.2.4.1

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

$x = \sqrt{5}$

Step 2.5.2.4.2

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

$x = - \sqrt{5}$

Step 2.5.2.4.3

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

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

Step 2.6

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

$x = 0, \sqrt{5}, - \sqrt{5}$

Step 3

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

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Step 3.1

Substitute $0$ in $f (x) = - \frac{2}{5} x^{6} + 5 x^{4}$ to find the value of $y$ .

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Step 3.1.1

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

$f (0) = - \frac{2}{5} \cdot {(0)}^{6} + 5 {(0)}^{4}$

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 $0$ .

$f (0) = - \frac{2}{5} \cdot 0 + 5 {(0)}^{4}$

Step 3.1.2.1.2

Multiply $- \frac{2}{5} \cdot 0$ .

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Step 3.1.2.1.2.1

Multiply $0$ by $- 1$ .

$f (0) = 0 (\frac{2}{5}) + 5 {(0)}^{4}$

Step 3.1.2.1.2.2

Multiply $0$ by $\frac{2}{5}$ .

$f (0) = 0 + 5 {(0)}^{4}$

Step 3.1.2.1.3

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

$f (0) = 0 + 5 \cdot 0$

Step 3.1.2.1.4

Multiply $5$ 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 $f (x) = - \frac{2}{5} x^{6} + 5 x^{4}$ is $(0, 0)$ . This point can be an inflection point.

$(0, 0)$

Step 3.3

Substitute $\sqrt{5}$ in $f (x) = - \frac{2}{5} x^{6} + 5 x^{4}$ to find the value of $y$ .

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Step 3.3.1

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

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

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 ${\sqrt{5}}^{6}$ as $5^{3}$ .

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Step 3.3.2.1.1.1

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

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

Step 3.3.2.1.1.2

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

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

Step 3.3.2.1.1.3

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

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

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 $6$ .

$f (\sqrt{5}) = - \frac{2}{5} \cdot 5^{\frac{2 \cdot 3}{2}} + 5 {(\sqrt{5})}^{4}$

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 $2$ .

$f (\sqrt{5}) = - \frac{2}{5} \cdot 5^{\frac{2 \cdot 3}{2 (1)}} + 5 {(\sqrt{5})}^{4}$

Step 3.3.2.1.1.4.2.2

Cancel the common factor.

$f (\sqrt{5}) = - \frac{2}{5} \cdot 5^{\frac{2 \cdot 3}{2 \cdot 1}} + 5 {(\sqrt{5})}^{4}$

Step 3.3.2.1.1.4.2.3

Rewrite the expression.

$f (\sqrt{5}) = - \frac{2}{5} \cdot 5^{\frac{3}{1}} + 5 {(\sqrt{5})}^{4}$

Step 3.3.2.1.1.4.2.4

Divide $3$ by $1$ .

$f (\sqrt{5}) = - \frac{2}{5} \cdot 5^{3} + 5 {(\sqrt{5})}^{4}$

Step 3.3.2.1.2

Cancel the common factor of $5$ .

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Step 3.3.2.1.2.1

Move the leading negative in $- \frac{2}{5}$ into the numerator.

$f (\sqrt{5}) = \frac{- 2}{5} \cdot 5^{3} + 5 {(\sqrt{5})}^{4}$

Step 3.3.2.1.2.2

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

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

Step 3.3.2.1.2.3

Cancel the common factor.

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

Step 3.3.2.1.2.4

Rewrite the expression.

$f (\sqrt{5}) = - 2 \cdot 5^{2} + 5 {(\sqrt{5})}^{4}$

Step 3.3.2.1.3

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

$f (\sqrt{5}) = - 2 \cdot 25 + 5 {(\sqrt{5})}^{4}$

Step 3.3.2.1.4

Multiply $- 2$ by $25$ .

$f (\sqrt{5}) = - 50 + 5 {(\sqrt{5})}^{4}$

Step 3.3.2.1.5

Rewrite ${\sqrt{5}}^{4}$ as $5^{2}$ .

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Step 3.3.2.1.5.1

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

$f (\sqrt{5}) = - 50 + 5 {(5^{\frac{1}{2}})}^{4}$

Step 3.3.2.1.5.2

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

$f (\sqrt{5}) = - 50 + 5 \cdot 5^{\frac{1}{2} \cdot 4}$

Step 3.3.2.1.5.3

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

$f (\sqrt{5}) = - 50 + 5 \cdot 5^{\frac{4}{2}}$

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 $4$ .

$f (\sqrt{5}) = - 50 + 5 \cdot 5^{\frac{2 \cdot 2}{2}}$

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 $2$ .

$f (\sqrt{5}) = - 50 + 5 \cdot 5^{\frac{2 \cdot 2}{2 (1)}}$

Step 3.3.2.1.5.4.2.2

Cancel the common factor.

$f (\sqrt{5}) = - 50 + 5 \cdot 5^{\frac{2 \cdot 2}{2 \cdot 1}}$

Step 3.3.2.1.5.4.2.3

Rewrite the expression.

$f (\sqrt{5}) = - 50 + 5 \cdot 5^{\frac{2}{1}}$

Step 3.3.2.1.5.4.2.4

Divide $2$ by $1$ .

$f (\sqrt{5}) = - 50 + 5 \cdot 5^{2}$

Step 3.3.2.1.6

Multiply $5$ by $5^{2}$ by adding the exponents.

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Step 3.3.2.1.6.1

Multiply $5$ by $5^{2}$ .

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Step 3.3.2.1.6.1.1

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

$f (\sqrt{5}) = - 50 + 5 \cdot 5^{2}$

Step 3.3.2.1.6.1.2

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

$f (\sqrt{5}) = - 50 + 5^{1 + 2}$

Step 3.3.2.1.6.2

Add $1$ and $2$ .

$f (\sqrt{5}) = - 50 + 5^{3}$

Step 3.3.2.1.7

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

$f (\sqrt{5}) = - 50 + 125$

Step 3.3.2.2

Add $- 50$ and $125$ .

$f (\sqrt{5}) = 75$

Step 3.3.2.3

The final answer is $75$ .

$75$

Step 3.4

The point found by substituting $\sqrt{5}$ in $f (x) = - \frac{2}{5} x^{6} + 5 x^{4}$ is $(\sqrt{5}, 75)$ . This point can be an inflection point.

$(\sqrt{5}, 75)$

Step 3.5

Substitute $- \sqrt{5}$ in $f (x) = - \frac{2}{5} x^{6} + 5 x^{4}$ to find the value of $y$ .

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Step 3.5.1

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

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

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 $- \sqrt{5}$ .

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

Step 3.5.2.1.2

Multiply $- 1$ by ${(- 1)}^{6}$ by adding the exponents.

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Step 3.5.2.1.2.1

Move ${(- 1)}^{6}$ .

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

Step 3.5.2.1.2.2

Multiply ${(- 1)}^{6}$ by $- 1$ .

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Step 3.5.2.1.2.2.1

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

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

Step 3.5.2.1.2.2.2

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

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

Step 3.5.2.1.2.3

Add $6$ and $1$ .

$f (- \sqrt{5}) = {(- 1)}^{7} (\frac{2}{5}) \cdot {\sqrt{5}}^{6} + 5 {(- \sqrt{5})}^{4}$

Step 3.5.2.1.3

Raise $- 1$ to the power of $7$ .

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

Step 3.5.2.1.4

Rewrite ${\sqrt{5}}^{6}$ as $5^{3}$ .

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Step 3.5.2.1.4.1

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

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

Step 3.5.2.1.4.2

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

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

Step 3.5.2.1.4.3

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

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

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 $6$ .

$f (- \sqrt{5}) = - \frac{2}{5} \cdot 5^{\frac{2 \cdot 3}{2}} + 5 {(- \sqrt{5})}^{4}$

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 $2$ .

$f (- \sqrt{5}) = - \frac{2}{5} \cdot 5^{\frac{2 \cdot 3}{2 (1)}} + 5 {(- \sqrt{5})}^{4}$

Step 3.5.2.1.4.4.2.2

Cancel the common factor.

$f (- \sqrt{5}) = - \frac{2}{5} \cdot 5^{\frac{2 \cdot 3}{2 \cdot 1}} + 5 {(- \sqrt{5})}^{4}$

Step 3.5.2.1.4.4.2.3

Rewrite the expression.

$f (- \sqrt{5}) = - \frac{2}{5} \cdot 5^{\frac{3}{1}} + 5 {(- \sqrt{5})}^{4}$

Step 3.5.2.1.4.4.2.4

Divide $3$ by $1$ .

$f (- \sqrt{5}) = - \frac{2}{5} \cdot 5^{3} + 5 {(- \sqrt{5})}^{4}$

Step 3.5.2.1.5

Cancel the common factor of $5$ .

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Step 3.5.2.1.5.1

Move the leading negative in $- \frac{2}{5}$ into the numerator.

$f (- \sqrt{5}) = \frac{- 2}{5} \cdot 5^{3} + 5 {(- \sqrt{5})}^{4}$

Step 3.5.2.1.5.2

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

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

Step 3.5.2.1.5.3

Cancel the common factor.

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

Step 3.5.2.1.5.4

Rewrite the expression.

$f (- \sqrt{5}) = - 2 \cdot 5^{2} + 5 {(- \sqrt{5})}^{4}$

Step 3.5.2.1.6

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

$f (- \sqrt{5}) = - 2 \cdot 25 + 5 {(- \sqrt{5})}^{4}$

Step 3.5.2.1.7

Multiply $- 2$ by $25$ .

$f (- \sqrt{5}) = - 50 + 5 {(- \sqrt{5})}^{4}$

Step 3.5.2.1.8

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

$f (- \sqrt{5}) = - 50 + 5 ({(- 1)}^{4} {\sqrt{5}}^{4})$

Step 3.5.2.1.9

Raise $- 1$ to the power of $4$ .

$f (- \sqrt{5}) = - 50 + 5 (1 {\sqrt{5}}^{4})$

Step 3.5.2.1.10

Multiply ${\sqrt{5}}^{4}$ by $1$ .

$f (- \sqrt{5}) = - 50 + 5 {\sqrt{5}}^{4}$

Step 3.5.2.1.11

Rewrite ${\sqrt{5}}^{4}$ as $5^{2}$ .

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Step 3.5.2.1.11.1

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

$f (- \sqrt{5}) = - 50 + 5 {(5^{\frac{1}{2}})}^{4}$

Step 3.5.2.1.11.2

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

$f (- \sqrt{5}) = - 50 + 5 \cdot 5^{\frac{1}{2} \cdot 4}$

Step 3.5.2.1.11.3

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

$f (- \sqrt{5}) = - 50 + 5 \cdot 5^{\frac{4}{2}}$

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 $4$ .

$f (- \sqrt{5}) = - 50 + 5 \cdot 5^{\frac{2 \cdot 2}{2}}$

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 $2$ .

$f (- \sqrt{5}) = - 50 + 5 \cdot 5^{\frac{2 \cdot 2}{2 (1)}}$

Step 3.5.2.1.11.4.2.2

Cancel the common factor.

$f (- \sqrt{5}) = - 50 + 5 \cdot 5^{\frac{2 \cdot 2}{2 \cdot 1}}$

Step 3.5.2.1.11.4.2.3

Rewrite the expression.

$f (- \sqrt{5}) = - 50 + 5 \cdot 5^{\frac{2}{1}}$

Step 3.5.2.1.11.4.2.4

Divide $2$ by $1$ .

$f (- \sqrt{5}) = - 50 + 5 \cdot 5^{2}$

Step 3.5.2.1.12

Multiply $5$ by $5^{2}$ by adding the exponents.

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Step 3.5.2.1.12.1

Multiply $5$ by $5^{2}$ .

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Step 3.5.2.1.12.1.1

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

$f (- \sqrt{5}) = - 50 + 5 \cdot 5^{2}$

Step 3.5.2.1.12.1.2

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

$f (- \sqrt{5}) = - 50 + 5^{1 + 2}$

Step 3.5.2.1.12.2

Add $1$ and $2$ .

$f (- \sqrt{5}) = - 50 + 5^{3}$

Step 3.5.2.1.13

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

$f (- \sqrt{5}) = - 50 + 125$

Step 3.5.2.2

Add $- 50$ and $125$ .

$f (- \sqrt{5}) = 75$

Step 3.5.2.3

The final answer is $75$ .

$75$

Step 3.6

The point found by substituting $- \sqrt{5}$ in $f (x) = - \frac{2}{5} x^{6} + 5 x^{4}$ is $(- \sqrt{5}, 75)$ . This point can be an inflection point.

$(- \sqrt{5}, 75)$

Step 3.7

Determine the points that could be inflection points.

$(0, 0), (\sqrt{5}, 75), (- \sqrt{5}, 75)$

Step 4

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

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

Step 5

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

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Step 5.1

Replace the variable $x$ with $- 2.33606797$ in the expression.

$f'' (- 2.33606797) = - 12 {(- 2.33606797)}^{4} + 60 {(- 2.33606797)}^{2}$

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 $- 2.33606797$ to the power of $4$ .

$f'' (- 2.33606797) = - 12 \cdot 29.78118022 + 60 {(- 2.33606797)}^{2}$

Step 5.2.1.2

Multiply $- 12$ by $29.78118022$ .

$f'' (- 2.33606797) = - 357.37416272 + 60 {(- 2.33606797)}^{2}$

Step 5.2.1.3

Raise $- 2.33606797$ to the power of $2$ .

$f'' (- 2.33606797) = - 357.37416272 + 60 \cdot 5.45721359$

Step 5.2.1.4

Multiply $60$ by $5.45721359$ .

$f'' (- 2.33606797) = - 357.37416272 + 327.43281572$

Step 5.2.2

Add $- 357.37416272$ and $327.43281572$ .

$f'' (- 2.33606797) = - 29.94134699$

Step 5.2.3

The final answer is $- 29.94134699$ .

$- 29.94134699$

Step 5.3

At $- 2.33606797$ , the second derivative is $- 29.94134699$ . Since this is negative, the second derivative is decreasing on the interval $(- \infty, - \sqrt{5})$

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

Step 6

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

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Step 6.1

Replace the variable $x$ with $- 1.11803398$ in the expression.

$f'' (- 1.11803398) = - 12 {(- 1.11803398)}^{4} + 60 {(- 1.11803398)}^{2}$

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 $- 1.11803398$ to the power of $4$ .

$f'' (- 1.11803398) = - 12 \cdot 1.5625 + 60 {(- 1.11803398)}^{2}$

Step 6.2.1.2

Multiply $- 12$ by $1.5625$ .

$f'' (- 1.11803398) = - 18.75 + 60 {(- 1.11803398)}^{2}$

Step 6.2.1.3

Raise $- 1.11803398$ to the power of $2$ .

$f'' (- 1.11803398) = - 18.75 + 60 \cdot 1.25$

Step 6.2.1.4

Multiply $60$ by $1.25$ .

$f'' (- 1.11803398) = - 18.75 + 75$

Step 6.2.2

Add $- 18.75$ and $75$ .

$f'' (- 1.11803398) = 56.25$

Step 6.2.3

The final answer is $56.25$ .

$56.25$

Step 6.3

At $- 1.11803398$ , the second derivative is $56.25$ . Since this is positive, the second derivative is increasing on the interval $(- \sqrt{5}, 0)$ .

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

Step 7

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

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Step 7.1

Replace the variable $x$ with $1.11803398$ in the expression.

$f'' (1.11803398) = - 12 {(1.11803398)}^{4} + 60 {(1.11803398)}^{2}$

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 $1.11803398$ to the power of $4$ .

$f'' (1.11803398) = - 12 \cdot 1.5625 + 60 {(1.11803398)}^{2}$

Step 7.2.1.2

Multiply $- 12$ by $1.5625$ .

$f'' (1.11803398) = - 18.75 + 60 {(1.11803398)}^{2}$

Step 7.2.1.3

Raise $1.11803398$ to the power of $2$ .

$f'' (1.11803398) = - 18.75 + 60 \cdot 1.25$

Step 7.2.1.4

Multiply $60$ by $1.25$ .

$f'' (1.11803398) = - 18.75 + 75$

Step 7.2.2

Add $- 18.75$ and $75$ .

$f'' (1.11803398) = 56.25$

Step 7.2.3

The final answer is $56.25$ .

$56.25$

Step 7.3

At $1.11803398$ , the second derivative is $56.25$ . Since this is positive, the second derivative is increasing on the interval $(0, \sqrt{5})$ .

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

Step 8

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

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Step 8.1

Replace the variable $x$ with $2.33606797$ in the expression.

$f'' (2.33606797) = - 12 {(2.33606797)}^{4} + 60 {(2.33606797)}^{2}$

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 $2.33606797$ to the power of $4$ .

$f'' (2.33606797) = - 12 \cdot 29.78118022 + 60 {(2.33606797)}^{2}$

Step 8.2.1.2

Multiply $- 12$ by $29.78118022$ .

$f'' (2.33606797) = - 357.37416272 + 60 {(2.33606797)}^{2}$

Step 8.2.1.3

Raise $2.33606797$ to the power of $2$ .

$f'' (2.33606797) = - 357.37416272 + 60 \cdot 5.45721359$

Step 8.2.1.4

Multiply $60$ by $5.45721359$ .

$f'' (2.33606797) = - 357.37416272 + 327.43281572$

Step 8.2.2

Add $- 357.37416272$ and $327.43281572$ .

$f'' (2.33606797) = - 29.94134699$

Step 8.2.3

The final answer is $- 29.94134699$ .

$- 29.94134699$

Step 8.3

At $2.33606797$ , the second derivative is $- 29.94134699$ . Since this is negative, the second derivative is decreasing on the interval $(\sqrt{5}, \infty)$

Decreasing on $(\sqrt{5}, \infty)$ 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 $(- \sqrt{5}, 75), (\sqrt{5}, 75)$ .

$(- \sqrt{5}, 75), (\sqrt{5}, 75)$

Step 10