Calculus Examples

$f (x) = \frac{2 x^{\frac{5}{2}}}{5} - \frac{4 x^{\frac{3}{2}}}{3} - \frac{x^{2}}{2} + 5$ , $[0, 5]$

Step 1

Find the critical points.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Find the first derivative.

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Step 1.1.1.1

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

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

Step 1.1.1.2

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

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Step 1.1.1.2.1

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

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

Step 1.1.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{5}{2}$ .

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

Step 1.1.1.2.3

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

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

Step 1.1.1.2.4

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

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

Step 1.1.1.2.5

Combine the numerators over the common denominator.

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

Step 1.1.1.2.6

Simplify the numerator.

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Step 1.1.1.2.6.1

Multiply $- 1$ by $2$ .

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

Step 1.1.1.2.6.2

Subtract $2$ from $5$ .

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

Step 1.1.1.2.7

Combine $\frac{5}{2}$ and $x^{\frac{3}{2}}$ .

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

Step 1.1.1.2.8

Multiply $\frac{2}{5}$ by $\frac{5 x^{\frac{3}{2}}}{2}$ .

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

Step 1.1.1.2.9

Multiply $5$ by $2$ .

$\frac{10 x^{\frac{3}{2}}}{5 \cdot 2} + \frac{d}{d x} [- \frac{4 x^{\frac{3}{2}}}{3}] + \frac{d}{d x} [- \frac{x^{2}}{2}] + \frac{d}{d x} [5]$

Step 1.1.1.2.10

Multiply $5$ by $2$ .

$\frac{10 x^{\frac{3}{2}}}{10} + \frac{d}{d x} [- \frac{4 x^{\frac{3}{2}}}{3}] + \frac{d}{d x} [- \frac{x^{2}}{2}] + \frac{d}{d x} [5]$

Step 1.1.1.2.11

Cancel the common factor.

$\frac{10 x^{\frac{3}{2}}}{10} + \frac{d}{d x} [- \frac{4 x^{\frac{3}{2}}}{3}] + \frac{d}{d x} [- \frac{x^{2}}{2}] + \frac{d}{d x} [5]$

Step 1.1.1.2.12

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

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

Step 1.1.1.3

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

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Step 1.1.1.3.1

Since $- \frac{4}{3}$ is constant with respect to $x$ , the derivative of $- \frac{4 x^{\frac{3}{2}}}{3}$ with respect to $x$ is $- \frac{4}{3} \frac{d}{d x} [x^{\frac{3}{2}}]$ .

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

Step 1.1.1.3.2

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

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

Step 1.1.1.3.3

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

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

Step 1.1.1.3.4

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

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

Step 1.1.1.3.5

Combine the numerators over the common denominator.

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

Step 1.1.1.3.6

Simplify the numerator.

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Step 1.1.1.3.6.1

Multiply $- 1$ by $2$ .

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

Step 1.1.1.3.6.2

Subtract $2$ from $3$ .

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

Step 1.1.1.3.7

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

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

Step 1.1.1.3.8

Multiply $\frac{3 x^{\frac{1}{2}}}{2}$ by $\frac{4}{3}$ .

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

Step 1.1.1.3.9

Multiply $4$ by $3$ .

$x^{\frac{3}{2}} - \frac{12 x^{\frac{1}{2}}}{2 \cdot 3} + \frac{d}{d x} [- \frac{x^{2}}{2}] + \frac{d}{d x} [5]$

Step 1.1.1.3.10

Multiply $2$ by $3$ .

$x^{\frac{3}{2}} - \frac{12 x^{\frac{1}{2}}}{6} + \frac{d}{d x} [- \frac{x^{2}}{2}] + \frac{d}{d x} [5]$

Step 1.1.1.3.11

Factor $6$ out of $12 x^{\frac{1}{2}}$ .

$x^{\frac{3}{2}} - \frac{6 (2 x^{\frac{1}{2}})}{6} + \frac{d}{d x} [- \frac{x^{2}}{2}] + \frac{d}{d x} [5]$

Step 1.1.1.3.12

Cancel the common factors.

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Step 1.1.1.3.12.1

Factor $6$ out of $6$ .

$x^{\frac{3}{2}} - \frac{6 (2 x^{\frac{1}{2}})}{6 (1)} + \frac{d}{d x} [- \frac{x^{2}}{2}] + \frac{d}{d x} [5]$

Step 1.1.1.3.12.2

Cancel the common factor.

$x^{\frac{3}{2}} - \frac{6 (2 x^{\frac{1}{2}})}{6 \cdot 1} + \frac{d}{d x} [- \frac{x^{2}}{2}] + \frac{d}{d x} [5]$

Step 1.1.1.3.12.3

Rewrite the expression.

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

Step 1.1.1.3.12.4

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

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

Step 1.1.1.3.13

Multiply $2$ by $- 1$ .

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

Step 1.1.1.4

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

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Step 1.1.1.4.1

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

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

Step 1.1.1.4.2

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

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

Step 1.1.1.4.3

Multiply $2$ by $- 1$ .

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

Step 1.1.1.4.4

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

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

Step 1.1.1.4.5

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

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

Step 1.1.1.4.6

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

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Step 1.1.1.4.6.1

Factor $2$ out of $- 2 x$ .

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

Step 1.1.1.4.6.2

Cancel the common factors.

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Step 1.1.1.4.6.2.1

Factor $2$ out of $2$ .

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

Step 1.1.1.4.6.2.2

Cancel the common factor.

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

Step 1.1.1.4.6.2.3

Rewrite the expression.

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

Step 1.1.1.4.6.2.4

Divide $- x$ by $1$ .

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

Step 1.1.1.5

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

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

Step 1.1.1.6

Simplify.

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Step 1.1.1.6.1

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

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

Step 1.1.1.6.2

Reorder terms.

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

Step 1.1.2

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

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

Step 1.2

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

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Step 1.2.1

Set the first derivative equal to $0$ .

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

Step 1.2.2

Find a common factor $x^{\frac{1}{2}}$ that is present in each term.

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

Step 1.2.3

Substitute $u$ for $x^{\frac{1}{2}}$ .

$- 1 {(u)}^{2} {(u)}^{3} - 2 u = 0$

Step 1.2.4

Solve for $u$ .

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Step 1.2.4.1

Simplify each term.

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Step 1.2.4.1.1

Multiply ${(u)}^{2}$ by ${(u)}^{3}$ by adding the exponents.

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Step 1.2.4.1.1.1

Move ${(u)}^{3}$ .

$- 1 ({(u)}^{3} {(u)}^{2}) - 2 u = 0$

Step 1.2.4.1.1.2

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

$- 1 u^{3 + 2} - 2 u = 0$

Step 1.2.4.1.1.3

Add $3$ and $2$ .

$- 1 u^{5} - 2 u = 0$

Step 1.2.4.1.2

Rewrite $- 1 u^{5}$ as $- u^{5}$ .

$- u^{5} - 2 u = 0$

Step 1.2.4.2

Factor $- u$ out of $- u^{5} - 2 u$ .

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Step 1.2.4.2.1

Factor $- u$ out of $- u^{5}$ .

$- u \cdot u^{4} - 2 u = 0$

Step 1.2.4.2.2

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

$- u \cdot u^{4} - u \cdot 2 = 0$

Step 1.2.4.2.3

Factor $- u$ out of $- u (u^{4}) - u (2)$ .

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

Step 1.2.4.3

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

$u = 0$

$u^{4} + 2 = 0$

Step 1.2.4.4

Set $u$ equal to $0$ .

$u = 0$

Step 1.2.4.5

Set $u^{4} + 2$ equal to $0$ and solve for $u$ .

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Step 1.2.4.5.1

Set $u^{4} + 2$ equal to $0$ .

$u^{4} + 2 = 0$

Step 1.2.4.5.2

Solve $u^{4} + 2 = 0$ for $u$ .

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Step 1.2.4.5.2.1

Subtract $2$ from both sides of the equation.

$u^{4} = - 2$

Step 1.2.4.5.2.2

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

$u = \pm \sqrt[4]{- 2}$

Step 1.2.4.5.2.3

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

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Step 1.2.4.5.2.3.1

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

$u = \sqrt[4]{- 2}$

Step 1.2.4.5.2.3.2

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

$u = - \sqrt[4]{- 2}$

Step 1.2.4.5.2.3.3

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

$u = \sqrt[4]{- 2}, - \sqrt[4]{- 2}$

Step 1.2.4.6

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

$u = 0, \sqrt[4]{- 2}, - \sqrt[4]{- 2}$

Step 1.2.5

Substitute $x$ for $u$ .

$x^{\frac{1}{2}} = 0, \sqrt[4]{- 2}, - \sqrt[4]{- 2}$

Step 1.2.6

Solve for $x^{\frac{1}{2}} = 0$ for $x$ .

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Step 1.2.6.1

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

${(x^{\frac{1}{2}})}^{2} = 0^{2}$

Step 1.2.6.2

Simplify the exponent.

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Step 1.2.6.2.1

Simplify the left side.

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Step 1.2.6.2.1.1

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

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Step 1.2.6.2.1.1.1

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

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Step 1.2.6.2.1.1.1.1

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

$x^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 1.2.6.2.1.1.1.2

Cancel the common factor of $2$ .

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Step 1.2.6.2.1.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 1.2.6.2.1.1.1.2.2

Rewrite the expression.

$x^{1} = 0^{2}$

Step 1.2.6.2.1.1.2

Simplify.

$x = 0^{2}$

Step 1.2.6.2.2

Simplify the right side.

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Step 1.2.6.2.2.1

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

$x = 0$

Step 1.2.7

Solve for $x^{\frac{1}{2}} = \sqrt[4]{- 2}$ for $x$ .

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Step 1.2.7.1

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

${(x^{\frac{1}{2}})}^{2} = {\sqrt[4]{- 2}}^{2}$

Step 1.2.7.2

Simplify the exponent.

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Step 1.2.7.2.1

Simplify the left side.

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Step 1.2.7.2.1.1

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

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Step 1.2.7.2.1.1.1

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

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Step 1.2.7.2.1.1.1.1

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

$x^{\frac{1}{2} \cdot 2} = {\sqrt[4]{- 2}}^{2}$

Step 1.2.7.2.1.1.1.2

Cancel the common factor of $2$ .

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Step 1.2.7.2.1.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = {\sqrt[4]{- 2}}^{2}$

Step 1.2.7.2.1.1.1.2.2

Rewrite the expression.

$x^{1} = {\sqrt[4]{- 2}}^{2}$

Step 1.2.7.2.1.1.2

Simplify.

$x = {\sqrt[4]{- 2}}^{2}$

Step 1.2.7.2.2

Simplify the right side.

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Step 1.2.7.2.2.1

Simplify ${\sqrt[4]{- 2}}^{2}$ .

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Step 1.2.7.2.2.1.1

Rewrite ${\sqrt[4]{- 2}}^{2}$ as $\sqrt{- 2}$ .

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Step 1.2.7.2.2.1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{- 2}$ as ${(- 2)}^{\frac{1}{4}}$ .

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

Step 1.2.7.2.2.1.1.2

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

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

Step 1.2.7.2.2.1.1.3

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

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

Step 1.2.7.2.2.1.1.4

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

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Step 1.2.7.2.2.1.1.4.1

Factor $2$ out of $2$ .

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

Step 1.2.7.2.2.1.1.4.2

Cancel the common factors.

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Step 1.2.7.2.2.1.1.4.2.1

Factor $2$ out of $4$ .

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

Step 1.2.7.2.2.1.1.4.2.2

Cancel the common factor.

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

Step 1.2.7.2.2.1.1.4.2.3

Rewrite the expression.

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

Step 1.2.7.2.2.1.1.5

Rewrite ${(- 2)}^{\frac{1}{2}}$ as $\sqrt{- 2}$ .

$x = \sqrt{- 2}$

Step 1.2.7.2.2.1.2

Rewrite $- 2$ as $- 1 (2)$ .

$x = \sqrt{- 1 (2)}$

Step 1.2.7.2.2.1.3

Rewrite $\sqrt{- 1 (2)}$ as $\sqrt{- 1} \cdot \sqrt{2}$ .

$x = \sqrt{- 1} \cdot \sqrt{2}$

Step 1.2.7.2.2.1.4

Rewrite $\sqrt{- 1}$ as $i$ .

$x = i \sqrt{2}$

Step 1.2.8

List all of the solutions.

$x = 0, i \sqrt{2}, i \sqrt{2}$

Step 1.3

Find the values where the derivative is undefined.

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Step 1.3.1

Convert expressions with fractional exponents to radicals.

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Step 1.3.1.1

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

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

Step 1.3.1.2

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

$- x + \sqrt{x^{3}} - 2 \sqrt{x^{1}}$

Step 1.3.1.3

Anything raised to $1$ is the base itself.

$- x + \sqrt{x^{3}} - 2 \sqrt{x}$

Step 1.3.2

Set the radicand in $\sqrt{x^{3}}$ less than $0$ to find where the expression is undefined.

$x^{3} < 0$

Step 1.3.3

Solve for $x$ .

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Step 1.3.3.1

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

$\sqrt[3]{x^{3}} < \sqrt[3]{0}$

Step 1.3.3.2

Simplify the equation.

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Step 1.3.3.2.1

Simplify the left side.

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Step 1.3.3.2.1.1

Pull terms out from under the radical.

$x < \sqrt[3]{0}$

Step 1.3.3.2.2

Simplify the right side.

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Step 1.3.3.2.2.1

Simplify $\sqrt[3]{0}$ .

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Step 1.3.3.2.2.1.1

Rewrite $0$ as $0^{3}$ .

$x < \sqrt[3]{0^{3}}$

Step 1.3.3.2.2.1.2

Pull terms out from under the radical.

$x < 0$

Step 1.3.4

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x < 0$

$(- \infty, 0)$

$x < 0$

$(- \infty, 0)$

Step 1.4

Evaluate $\frac{2 x^{\frac{5}{2}}}{5} - \frac{4 x^{\frac{3}{2}}}{3} - \frac{x^{2}}{2} + 5$ at each $x$ value where the derivative is $0$ or undefined.

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Step 1.4.1

Evaluate at $x = 0$ .

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Step 1.4.1.1

Substitute $0$ for $x$ .

$\frac{2 {(0)}^{\frac{5}{2}}}{5} - \frac{4 {(0)}^{\frac{3}{2}}}{3} - \frac{{(0)}^{2}}{2} + 5$

Step 1.4.1.2

Simplify.

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Step 1.4.1.2.1

Simplify each term.

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Step 1.4.1.2.1.1

Simplify the numerator.

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Step 1.4.1.2.1.1.1

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

$\frac{2 \cdot {(0^{2})}^{\frac{5}{2}}}{5} - \frac{4 {(0)}^{\frac{3}{2}}}{3} - \frac{{(0)}^{2}}{2} + 5$

Step 1.4.1.2.1.1.2

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

$\frac{2 \cdot 0^{2 (\frac{5}{2})}}{5} - \frac{4 {(0)}^{\frac{3}{2}}}{3} - \frac{{(0)}^{2}}{2} + 5$

Step 1.4.1.2.1.1.3

Cancel the common factor of $2$ .

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Step 1.4.1.2.1.1.3.1

Cancel the common factor.

$\frac{2 \cdot 0^{2 (\frac{5}{2})}}{5} - \frac{4 {(0)}^{\frac{3}{2}}}{3} - \frac{{(0)}^{2}}{2} + 5$

Step 1.4.1.2.1.1.3.2

Rewrite the expression.

$\frac{2 \cdot 0^{5}}{5} - \frac{4 {(0)}^{\frac{3}{2}}}{3} - \frac{{(0)}^{2}}{2} + 5$

Step 1.4.1.2.1.1.4

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

$\frac{2 \cdot 0}{5} - \frac{4 {(0)}^{\frac{3}{2}}}{3} - \frac{{(0)}^{2}}{2} + 5$

Step 1.4.1.2.1.2

Multiply $2$ by $0$ .

$\frac{0}{5} - \frac{4 {(0)}^{\frac{3}{2}}}{3} - \frac{{(0)}^{2}}{2} + 5$

Step 1.4.1.2.1.3

Divide $0$ by $5$ .

$0 - \frac{4 {(0)}^{\frac{3}{2}}}{3} - \frac{{(0)}^{2}}{2} + 5$

Step 1.4.1.2.1.4

Simplify the numerator.

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Step 1.4.1.2.1.4.1

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

$0 - \frac{4 \cdot {(0^{2})}^{\frac{3}{2}}}{3} - \frac{{(0)}^{2}}{2} + 5$

Step 1.4.1.2.1.4.2

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

$0 - \frac{4 \cdot 0^{2 (\frac{3}{2})}}{3} - \frac{{(0)}^{2}}{2} + 5$

Step 1.4.1.2.1.4.3

Cancel the common factor of $2$ .

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Step 1.4.1.2.1.4.3.1

Cancel the common factor.

$0 - \frac{4 \cdot 0^{2 (\frac{3}{2})}}{3} - \frac{{(0)}^{2}}{2} + 5$

Step 1.4.1.2.1.4.3.2

Rewrite the expression.

$0 - \frac{4 \cdot 0^{3}}{3} - \frac{{(0)}^{2}}{2} + 5$

Step 1.4.1.2.1.4.4

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

$0 - \frac{4 \cdot 0}{3} - \frac{{(0)}^{2}}{2} + 5$

Step 1.4.1.2.1.5

Multiply $4$ by $0$ .

$0 - \frac{0}{3} - \frac{{(0)}^{2}}{2} + 5$

Step 1.4.1.2.1.6

Divide $0$ by $3$ .

$0 - 0 - \frac{{(0)}^{2}}{2} + 5$

Step 1.4.1.2.1.7

Multiply $- 1$ by $0$ .

$0 + 0 - \frac{{(0)}^{2}}{2} + 5$

Step 1.4.1.2.1.8

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

$0 + 0 - \frac{0}{2} + 5$

Step 1.4.1.2.1.9

Divide $0$ by $2$ .

$0 + 0 - 0 + 5$

Step 1.4.1.2.1.10

Multiply $- 1$ by $0$ .

$0 + 0 + 0 + 5$

Step 1.4.1.2.2

Simplify by adding numbers.

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Step 1.4.1.2.2.1

Add $0$ and $0$ .

$0 + 0 + 5$

Step 1.4.1.2.2.2

Add $0$ and $0$ .

$0 + 5$

Step 1.4.1.2.2.3

Add $0$ and $5$ .

$5$

Step 1.4.2

List all of the points.

$(0, 5)$

Step 2

Evaluate at the included endpoints.

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Step 2.1

Evaluate at $x = 0$ .

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Step 2.1.1

Substitute $0$ for $x$ .

$\frac{2 {(0)}^{\frac{5}{2}}}{5} - \frac{4 {(0)}^{\frac{3}{2}}}{3} - \frac{{(0)}^{2}}{2} + 5$

Step 2.1.2

Simplify.

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Step 2.1.2.1

Simplify each term.

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Step 2.1.2.1.1

Simplify the numerator.

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Step 2.1.2.1.1.1

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

$\frac{2 \cdot {(0^{2})}^{\frac{5}{2}}}{5} - \frac{4 {(0)}^{\frac{3}{2}}}{3} - \frac{{(0)}^{2}}{2} + 5$

Step 2.1.2.1.1.2

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

$\frac{2 \cdot 0^{2 (\frac{5}{2})}}{5} - \frac{4 {(0)}^{\frac{3}{2}}}{3} - \frac{{(0)}^{2}}{2} + 5$

Step 2.1.2.1.1.3

Cancel the common factor of $2$ .

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Step 2.1.2.1.1.3.1

Cancel the common factor.

$\frac{2 \cdot 0^{2 (\frac{5}{2})}}{5} - \frac{4 {(0)}^{\frac{3}{2}}}{3} - \frac{{(0)}^{2}}{2} + 5$

Step 2.1.2.1.1.3.2

Rewrite the expression.

$\frac{2 \cdot 0^{5}}{5} - \frac{4 {(0)}^{\frac{3}{2}}}{3} - \frac{{(0)}^{2}}{2} + 5$

Step 2.1.2.1.1.4

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

$\frac{2 \cdot 0}{5} - \frac{4 {(0)}^{\frac{3}{2}}}{3} - \frac{{(0)}^{2}}{2} + 5$

Step 2.1.2.1.2

Multiply $2$ by $0$ .

$\frac{0}{5} - \frac{4 {(0)}^{\frac{3}{2}}}{3} - \frac{{(0)}^{2}}{2} + 5$

Step 2.1.2.1.3

Divide $0$ by $5$ .

$0 - \frac{4 {(0)}^{\frac{3}{2}}}{3} - \frac{{(0)}^{2}}{2} + 5$

Step 2.1.2.1.4

Simplify the numerator.

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Step 2.1.2.1.4.1

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

$0 - \frac{4 \cdot {(0^{2})}^{\frac{3}{2}}}{3} - \frac{{(0)}^{2}}{2} + 5$

Step 2.1.2.1.4.2

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

$0 - \frac{4 \cdot 0^{2 (\frac{3}{2})}}{3} - \frac{{(0)}^{2}}{2} + 5$

Step 2.1.2.1.4.3

Cancel the common factor of $2$ .

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Step 2.1.2.1.4.3.1

Cancel the common factor.

$0 - \frac{4 \cdot 0^{2 (\frac{3}{2})}}{3} - \frac{{(0)}^{2}}{2} + 5$

Step 2.1.2.1.4.3.2

Rewrite the expression.

$0 - \frac{4 \cdot 0^{3}}{3} - \frac{{(0)}^{2}}{2} + 5$

Step 2.1.2.1.4.4

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

$0 - \frac{4 \cdot 0}{3} - \frac{{(0)}^{2}}{2} + 5$

Step 2.1.2.1.5

Multiply $4$ by $0$ .

$0 - \frac{0}{3} - \frac{{(0)}^{2}}{2} + 5$

Step 2.1.2.1.6

Divide $0$ by $3$ .

$0 - 0 - \frac{{(0)}^{2}}{2} + 5$

Step 2.1.2.1.7

Multiply $- 1$ by $0$ .

$0 + 0 - \frac{{(0)}^{2}}{2} + 5$

Step 2.1.2.1.8

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

$0 + 0 - \frac{0}{2} + 5$

Step 2.1.2.1.9

Divide $0$ by $2$ .

$0 + 0 - 0 + 5$

Step 2.1.2.1.10

Multiply $- 1$ by $0$ .

$0 + 0 + 0 + 5$

Step 2.1.2.2

Simplify by adding numbers.

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Step 2.1.2.2.1

Add $0$ and $0$ .

$0 + 0 + 5$

Step 2.1.2.2.2

Add $0$ and $0$ .

$0 + 5$

Step 2.1.2.2.3

Add $0$ and $5$ .

$5$

Step 2.2

Evaluate at $x = 5$ .

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Step 2.2.1

Substitute $5$ for $x$ .

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

Step 2.2.2

Simplify.

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Step 2.2.2.1

Simplify each term.

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Step 2.2.2.1.1

Move $5^{1}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

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

Step 2.2.2.1.2

Multiply ${(5)}^{\frac{5}{2}}$ by $5^{- 1}$ by adding the exponents.

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Step 2.2.2.1.2.1

Move $5^{- 1}$ .

$2 (5^{- 1} {(5)}^{\frac{5}{2}}) - \frac{4 {(5)}^{\frac{3}{2}}}{3} - \frac{{(5)}^{2}}{2} + 5$

Step 2.2.2.1.2.2

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

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

Step 2.2.2.1.2.3

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

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

Step 2.2.2.1.2.4

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

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

Step 2.2.2.1.2.5

Combine the numerators over the common denominator.

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

Step 2.2.2.1.2.6

Simplify the numerator.

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Step 2.2.2.1.2.6.1

Multiply $- 1$ by $2$ .

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

Step 2.2.2.1.2.6.2

Add $- 2$ and $5$ .

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

Step 2.2.2.1.3

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

$2 \cdot 5^{\frac{3}{2}} - \frac{4 \cdot 5^{\frac{3}{2}}}{3} - \frac{25}{2} + 5$

Step 2.2.2.2

To write $2 \cdot 5^{\frac{3}{2}}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$2 \cdot 5^{\frac{3}{2}} \cdot \frac{3}{3} - \frac{4 \cdot 5^{\frac{3}{2}}}{3} - \frac{25}{2} + 5$

Step 2.2.2.3

Combine fractions.

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Step 2.2.2.3.1

Combine $2 \cdot 5^{\frac{3}{2}}$ and $\frac{3}{3}$ .

$\frac{2 \cdot 5^{\frac{3}{2}} \cdot 3}{3} - \frac{4 \cdot 5^{\frac{3}{2}}}{3} - \frac{25}{2} + 5$

Step 2.2.2.3.2

Combine the numerators over the common denominator.

$\frac{2 \cdot 5^{\frac{3}{2}} \cdot 3 - 4 \cdot 5^{\frac{3}{2}}}{3} - \frac{25}{2} + 5$

Step 2.2.2.4

Simplify the numerator.

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Step 2.2.2.4.1

Multiply $3$ by $2$ .

$\frac{6 \cdot 5^{\frac{3}{2}} - 4 \cdot 5^{\frac{3}{2}}}{3} - \frac{25}{2} + 5$

Step 2.2.2.4.2

Subtract $4 \cdot 5^{\frac{3}{2}}$ from $6 \cdot 5^{\frac{3}{2}}$ .

$\frac{2 \cdot 5^{\frac{3}{2}}}{3} - \frac{25}{2} + 5$

Step 2.2.2.5

To write $5$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{2 \cdot 5^{\frac{3}{2}}}{3} + 5 \cdot \frac{3}{3} - \frac{25}{2}$

Step 2.2.2.6

Combine $5$ and $\frac{3}{3}$ .

$\frac{2 \cdot 5^{\frac{3}{2}}}{3} + \frac{5 \cdot 3}{3} - \frac{25}{2}$

Step 2.2.2.7

Simplify the expression.

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Step 2.2.2.7.1

Combine the numerators over the common denominator.

$\frac{2 \cdot 5^{\frac{3}{2}} + 5 \cdot 3}{3} - \frac{25}{2}$

Step 2.2.2.7.2

Multiply $5$ by $3$ .

$\frac{2 \cdot 5^{\frac{3}{2}} + 15}{3} - \frac{25}{2}$

Step 2.2.2.8

To write $\frac{2 \cdot 5^{\frac{3}{2}} + 15}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{2 \cdot 5^{\frac{3}{2}} + 15}{3} \cdot \frac{2}{2} - \frac{25}{2}$

Step 2.2.2.9

To write $- \frac{25}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{2 \cdot 5^{\frac{3}{2}} + 15}{3} \cdot \frac{2}{2} - \frac{25}{2} \cdot \frac{3}{3}$

Step 2.2.2.10

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.2.2.10.1

Multiply $\frac{2 \cdot 5^{\frac{3}{2}} + 15}{3}$ by $\frac{2}{2}$ .

$\frac{(2 \cdot 5^{\frac{3}{2}} + 15) \cdot 2}{3 \cdot 2} - \frac{25}{2} \cdot \frac{3}{3}$

Step 2.2.2.10.2

Multiply $3$ by $2$ .

$\frac{(2 \cdot 5^{\frac{3}{2}} + 15) \cdot 2}{6} - \frac{25}{2} \cdot \frac{3}{3}$

Step 2.2.2.10.3

Multiply $\frac{25}{2}$ by $\frac{3}{3}$ .

$\frac{(2 \cdot 5^{\frac{3}{2}} + 15) \cdot 2}{6} - \frac{25 \cdot 3}{2 \cdot 3}$

Step 2.2.2.10.4

Multiply $2$ by $3$ .

$\frac{(2 \cdot 5^{\frac{3}{2}} + 15) \cdot 2}{6} - \frac{25 \cdot 3}{6}$

Step 2.2.2.11

Combine the numerators over the common denominator.

$\frac{(2 \cdot 5^{\frac{3}{2}} + 15) \cdot 2 - 25 \cdot 3}{6}$

Step 2.2.2.12

Simplify the numerator.

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Step 2.2.2.12.1

Apply the distributive property.

$\frac{2 \cdot 5^{\frac{3}{2}} \cdot 2 + 15 \cdot 2 - 25 \cdot 3}{6}$

Step 2.2.2.12.2

Multiply $2$ by $2$ .

$\frac{4 \cdot 5^{\frac{3}{2}} + 15 \cdot 2 - 25 \cdot 3}{6}$

Step 2.2.2.12.3

Multiply $15$ by $2$ .

$\frac{4 \cdot 5^{\frac{3}{2}} + 30 - 25 \cdot 3}{6}$

Step 2.2.2.12.4

Multiply $- 25$ by $3$ .

$\frac{4 \cdot 5^{\frac{3}{2}} + 30 - 75}{6}$

Step 2.2.2.12.5

Subtract $75$ from $30$ .

$\frac{4 \cdot 5^{\frac{3}{2}} - 45}{6}$

Step 2.3

List all of the points.

$(0, 5), (5, \frac{4 \cdot 5^{\frac{3}{2}} - 45}{6})$

Step 3

Compare the $f (x)$ values found for each value of $x$ in order to determine the absolute maximum and minimum over the given interval. The maximum will occur at the highest $f (x)$ value and the minimum will occur at the lowest $f (x)$ value.

Absolute Maximum: $(0, 5)$

Absolute Minimum: $(5, \frac{4 \cdot 5^{\frac{3}{2}} - 45}{6})$

Step 4