Calculus Examples

f (x) = \frac{2 x^{\frac{5}{2}}}{5} - \frac{2 x^{\frac{3}{2}}}{3} - 6

$f (x) = \frac{2 x^{\frac{5}{2}}}{5} - \frac{2 x^{\frac{3}{2}}}{3} - 6$ ,

[0, 4]

$[0, 4]$

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{2 x^{\frac{3}{2}}}{3} - 6

$\frac{2 x^{\frac{5}{2}}}{5} - \frac{2 x^{\frac{3}{2}}}{3} - 6$ with respect to

x

$x$ is

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

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

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

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

Step 1.1.1.2

Evaluate

\frac{d}{d x} [\frac{2 x^{\frac{5}{2}}}{5}]

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

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Step 1.1.1.2.1

Since

\frac{2}{5}

$\frac{2}{5}$ is constant with respect to

x

$x$ , the derivative of

\frac{2 x^{\frac{5}{2}}}{5}

$\frac{2 x^{\frac{5}{2}}}{5}$ with respect to

x

$x$ is

\frac{2}{5} \frac{d}{d x} [x^{\frac{5}{2}}]

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

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

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

Step 1.1.1.2.2

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = \frac{5}{2}

$n = \frac{5}{2}$ .

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

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

Step 1.1.1.2.3

To write

- 1

$- 1$ as a fraction with a common denominator, multiply by

\frac{2}{2}

$\frac{2}{2}$ .

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

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

Step 1.1.1.2.4

Combine

- 1

$- 1$ and

\frac{2}{2}

$\frac{2}{2}$ .

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

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

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{2 x^{\frac{3}{2}}}{3}] + \frac{d}{d x} [- 6]

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

Step 1.1.1.2.6

Simplify the numerator.

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Step 1.1.1.2.6.1

Multiply

- 1

$- 1$ by

2

$2$ .

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

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

Step 1.1.1.2.6.2

Subtract

2

$2$ from

5

$5$ .

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

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

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

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

Step 1.1.1.2.7

Combine

\frac{5}{2}

$\frac{5}{2}$ and

x^{\frac{3}{2}}

$x^{\frac{3}{2}}$ .

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

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

Step 1.1.1.2.8

Multiply

\frac{2}{5}

$\frac{2}{5}$ by

\frac{5 x^{\frac{3}{2}}}{2}

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

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

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

Step 1.1.1.2.9

Multiply

5

$5$ by

2

$2$ .

\frac{10 x^{\frac{3}{2}}}{5 \cdot 2} + \frac{d}{d x} [- \frac{2 x^{\frac{3}{2}}}{3}] + \frac{d}{d x} [- 6]

$\frac{10 x^{\frac{3}{2}}}{5 \cdot 2} + \frac{d}{d x} [- \frac{2 x^{\frac{3}{2}}}{3}] + \frac{d}{d x} [- 6]$

Step 1.1.1.2.10

Multiply

5

$5$ by

2

$2$ .

\frac{10 x^{\frac{3}{2}}}{10} + \frac{d}{d x} [- \frac{2 x^{\frac{3}{2}}}{3}] + \frac{d}{d x} [- 6]

$\frac{10 x^{\frac{3}{2}}}{10} + \frac{d}{d x} [- \frac{2 x^{\frac{3}{2}}}{3}] + \frac{d}{d x} [- 6]$

Step 1.1.1.2.11

Cancel the common factor.

$\frac{10 x^{\frac{3}{2}}}{10} + \frac{d}{d x} [- \frac{2 x^{\frac{3}{2}}}{3}] + \frac{d}{d x} [- 6]$

Step 1.1.1.2.12

Divide

$x^{\frac{3}{2}}$ by

$1$ .

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

Step 1.1.1.3

Evaluate

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

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Step 1.1.1.3.1

Since

$- \frac{2}{3}$ is constant with respect to

$x$ , the derivative of

$- \frac{2 x^{\frac{3}{2}}}{3}$ with respect to

$x$ is

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

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

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

$n = \frac{3}{2}$ .

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

Step 1.1.1.3.3

To write

$- 1$ as a fraction with a common denominator, multiply by

$\frac{2}{2}$ .

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

Step 1.1.1.3.4

Combine

$- 1$ and

$\frac{2}{2}$ .

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

Step 1.1.1.3.5

Combine the numerators over the common denominator.

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

Step 1.1.1.3.6

Simplify the numerator.

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Step 1.1.1.3.6.1

Multiply

$- 1$ by

$2$ .

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

Step 1.1.1.3.6.2

Subtract

$2$ from

$3$ .

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

Step 1.1.1.3.7

Combine

$\frac{3}{2}$ and

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

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

Step 1.1.1.3.8

Multiply

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

$\frac{2}{3}$ .

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

Step 1.1.1.3.9

Multiply

$2$ by

$3$ .

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

Step 1.1.1.3.10

Multiply

$2$ by

$3$ .

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

Step 1.1.1.3.11

Cancel the common factor.

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

Step 1.1.1.3.12

Divide

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

$1$ .

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

Step 1.1.1.4

Differentiate using the Constant Rule.

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Step 1.1.1.4.1

Since

$- 6$ is constant with respect to

$x$ , the derivative of

$- 6$ with respect to

$x$ is

$0$ .

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

Step 1.1.1.4.2

Add

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

$0$ .

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

Step 1.1.2

The first derivative of

$f (x)$ with respect to

$x$ is

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

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

Step 1.2

Set the first derivative equal to

$0$ then solve the equation

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

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Step 1.2.1

Set the first derivative equal to

$0$ .

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

Step 1.2.2

Find a common factor

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

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

Step 1.2.3

Substitute

$u$ for

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

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

Step 1.2.4

Solve for

$u$ .

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Step 1.2.4.1

Multiply

$- 1$ by

$u$ .

$u^{3} - u = 0$

Step 1.2.4.2

Factor the left side of the equation.

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Step 1.2.4.2.1

Factor

$u$ out of

$u^{3} - u$ .

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Step 1.2.4.2.1.1

Factor

$u$ out of

$u^{3}$ .

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

Step 1.2.4.2.1.2

Factor

$u$ out of

$- u$ .

$u \cdot u^{2} + u \cdot - 1 = 0$

Step 1.2.4.2.1.3

Factor

$u$ out of

$u \cdot u^{2} + u \cdot - 1$ .

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

Step 1.2.4.2.2

Rewrite

$1$ as

$1^{2}$ .

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

Step 1.2.4.2.3

Factor.

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Step 1.2.4.2.3.1

Since both terms are perfect squares, factor using the difference of squares formula,

$a^{2} - b^{2} = (a + b) (a - b)$ where

$a = u$ and

$b = 1$ .

$u ((u + 1) (u - 1)) = 0$

Step 1.2.4.2.3.2

Remove unnecessary parentheses.

$u (u + 1) (u - 1) = 0$

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

$0$ .

$u = 0$

$u + 1 = 0$

$u - 1 = 0$

Step 1.2.4.4

Set

$u$ equal to

$0$ .

$u = 0$

Step 1.2.4.5

Set

$u + 1$ equal to

$0$ and solve for

$u$ .

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Step 1.2.4.5.1

Set

$u + 1$ equal to

$0$ .

$u + 1 = 0$

Step 1.2.4.5.2

Subtract

$1$ from both sides of the equation.

$u = - 1$

Step 1.2.4.6

Set

$u - 1$ equal to

$0$ and solve for

$u$ .

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Step 1.2.4.6.1

Set

$u - 1$ equal to

$0$ .

$u - 1 = 0$

Step 1.2.4.6.2

Add

$1$ to both sides of the equation.

$u = 1$

Step 1.2.4.7

The final solution is all the values that make

$u (u + 1) (u - 1) = 0$ true.

$u = 0, - 1, 1$

Step 1.2.5

Substitute

$x$ for

$u$ .

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

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

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

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

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,

${(a^{m})}^{n} = a^{m n}$ .

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

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.

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

Step 1.2.7.2.1.1.1.2.2

Rewrite the expression.

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

Step 1.2.7.2.1.1.2

Simplify.

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

Step 1.2.7.2.2

Simplify the right side.

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Step 1.2.7.2.2.1

Raise

$- 1$ to the power of

$2$ .

$x = 1$

Step 1.2.8

List all of the solutions.

$x = 0, 1, 1$

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.

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

Step 1.3.1.2

Apply the rule

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

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

Step 1.3.1.3

Anything raised to

$1$ is the base itself.

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

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

$\frac{2 x^{\frac{5}{2}}}{5} - \frac{2 x^{\frac{3}{2}}}{3} - 6$ 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

$x$ .

$\frac{2 {(0)}^{\frac{5}{2}}}{5} - \frac{2 {(0)}^{\frac{3}{2}}}{3} - 6$

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

$0^{2}$ .

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

Step 1.4.1.2.1.1.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

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

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.

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

Step 1.4.1.2.1.1.3.2

Rewrite the expression.

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

Step 1.4.1.2.1.1.4

Raising

$0$ to any positive power yields

$0$ .

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

Step 1.4.1.2.1.2

Multiply

$2$ by

$0$ .

$\frac{0}{5} - \frac{2 {(0)}^{\frac{3}{2}}}{3} - 6$

Step 1.4.1.2.1.3

Divide

$0$ by

$5$ .

$0 - \frac{2 {(0)}^{\frac{3}{2}}}{3} - 6$

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

$0^{2}$ .

$0 - \frac{2 \cdot {(0^{2})}^{\frac{3}{2}}}{3} - 6$

Step 1.4.1.2.1.4.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$0 - \frac{2 \cdot 0^{2 (\frac{3}{2})}}{3} - 6$

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.

$0 - \frac{2 \cdot 0^{2 (\frac{3}{2})}}{3} - 6$

Step 1.4.1.2.1.4.3.2

Rewrite the expression.

$0 - \frac{2 \cdot 0^{3}}{3} - 6$

Step 1.4.1.2.1.4.4

Raising

$0$ to any positive power yields

$0$ .

$0 - \frac{2 \cdot 0}{3} - 6$

Step 1.4.1.2.1.5

Multiply

$2$ by

$0$ .

$0 - \frac{0}{3} - 6$

Step 1.4.1.2.1.6

Divide

$0$ by

$3$ .

$0 - 0 - 6$

Step 1.4.1.2.1.7

Multiply

$- 1$ by

$0$ .

$0 + 0 - 6$

Step 1.4.1.2.2

Simplify by adding and subtracting.

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Step 1.4.1.2.2.1

Add

$0$ and

$0$ .

$0 - 6$

Step 1.4.1.2.2.2

Subtract

$6$ from

$0$ .

$- 6$

Step 1.4.2

Evaluate at

$x = 1$ .

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Step 1.4.2.1

Substitute

$1$ for

$x$ .

$\frac{2 {(1)}^{\frac{5}{2}}}{5} - \frac{2 {(1)}^{\frac{3}{2}}}{3} - 6$

Step 1.4.2.2

Simplify.

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Step 1.4.2.2.1

Simplify each term.

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Step 1.4.2.2.1.1

One to any power is one.

$\frac{2 \cdot 1}{5} - \frac{2 {(1)}^{\frac{3}{2}}}{3} - 6$

Step 1.4.2.2.1.2

Multiply

$2$ by

$1$ .

$\frac{2}{5} - \frac{2 {(1)}^{\frac{3}{2}}}{3} - 6$

Step 1.4.2.2.1.3

One to any power is one.

$\frac{2}{5} - \frac{2 \cdot 1}{3} - 6$

Step 1.4.2.2.1.4

Multiply

$2$ by

$1$ .

$\frac{2}{5} - \frac{2}{3} - 6$

Step 1.4.2.2.2

Find the common denominator.

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Step 1.4.2.2.2.1

Multiply

$\frac{2}{5}$ by

$\frac{3}{3}$ .

$\frac{2}{5} \cdot \frac{3}{3} - \frac{2}{3} - 6$

Step 1.4.2.2.2.2

Multiply

$\frac{2}{5}$ by

$\frac{3}{3}$ .

$\frac{2 \cdot 3}{5 \cdot 3} - \frac{2}{3} - 6$

Step 1.4.2.2.2.3

Multiply

$\frac{2}{3}$ by

$\frac{5}{5}$ .

$\frac{2 \cdot 3}{5 \cdot 3} - (\frac{2}{3} \cdot \frac{5}{5}) - 6$

Step 1.4.2.2.2.4

Multiply

$\frac{2}{3}$ by

$\frac{5}{5}$ .

$\frac{2 \cdot 3}{5 \cdot 3} - \frac{2 \cdot 5}{3 \cdot 5} - 6$

Step 1.4.2.2.2.5

Write

$- 6$ as a fraction with denominator

$1$ .

$\frac{2 \cdot 3}{5 \cdot 3} - \frac{2 \cdot 5}{3 \cdot 5} + \frac{- 6}{1}$

Step 1.4.2.2.2.6

Multiply

$\frac{- 6}{1}$ by

$\frac{15}{15}$ .

$\frac{2 \cdot 3}{5 \cdot 3} - \frac{2 \cdot 5}{3 \cdot 5} + \frac{- 6}{1} \cdot \frac{15}{15}$

Step 1.4.2.2.2.7

Multiply

$\frac{- 6}{1}$ by

$\frac{15}{15}$ .

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

Step 1.4.2.2.2.8

Reorder the factors of

$5 \cdot 3$ .

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

Step 1.4.2.2.2.9

Multiply

$3$ by

$5$ .

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

Step 1.4.2.2.2.10

Multiply

$3$ by

$5$ .

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

Step 1.4.2.2.3

Combine the numerators over the common denominator.

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

Step 1.4.2.2.4

Simplify each term.

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Step 1.4.2.2.4.1

Multiply

$2$ by

$3$ .

$\frac{6 - 2 \cdot 5 - 6 \cdot 15}{15}$

Step 1.4.2.2.4.2

Multiply

$- 2$ by

$5$ .

$\frac{6 - 10 - 6 \cdot 15}{15}$

Step 1.4.2.2.4.3

Multiply

$- 6$ by

$15$ .

$\frac{6 - 10 - 90}{15}$

Step 1.4.2.2.5

Simplify the expression.

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Step 1.4.2.2.5.1

Subtract

$10$ from

$6$ .

$\frac{- 4 - 90}{15}$

Step 1.4.2.2.5.2

Subtract

$90$ from

$- 4$ .

$\frac{- 94}{15}$

Step 1.4.2.2.5.3

Move the negative in front of the fraction.

$- \frac{94}{15}$

Step 1.4.3

List all of the points.

$(0, - 6), (1, - \frac{94}{15})$

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

$x$ .

$\frac{2 {(0)}^{\frac{5}{2}}}{5} - \frac{2 {(0)}^{\frac{3}{2}}}{3} - 6$

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

$0^{2}$ .

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

Step 2.1.2.1.1.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

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

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.

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

Step 2.1.2.1.1.3.2

Rewrite the expression.

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

Step 2.1.2.1.1.4

Raising

$0$ to any positive power yields

$0$ .

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

Step 2.1.2.1.2

Multiply

$2$ by

$0$ .

$\frac{0}{5} - \frac{2 {(0)}^{\frac{3}{2}}}{3} - 6$

Step 2.1.2.1.3

Divide

$0$ by

$5$ .

$0 - \frac{2 {(0)}^{\frac{3}{2}}}{3} - 6$

Step 2.1.2.1.4

Simplify the numerator.

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Step 2.1.2.1.4.1

Rewrite

$0$ as

$0^{2}$ .

$0 - \frac{2 \cdot {(0^{2})}^{\frac{3}{2}}}{3} - 6$

Step 2.1.2.1.4.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$0 - \frac{2 \cdot 0^{2 (\frac{3}{2})}}{3} - 6$

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.

$0 - \frac{2 \cdot 0^{2 (\frac{3}{2})}}{3} - 6$

Step 2.1.2.1.4.3.2

Rewrite the expression.

$0 - \frac{2 \cdot 0^{3}}{3} - 6$

Step 2.1.2.1.4.4

Raising

$0$ to any positive power yields

$0$ .

$0 - \frac{2 \cdot 0}{3} - 6$

Step 2.1.2.1.5

Multiply

$2$ by

$0$ .

$0 - \frac{0}{3} - 6$

Step 2.1.2.1.6

Divide

$0$ by

$3$ .

$0 - 0 - 6$

Step 2.1.2.1.7

Multiply

$- 1$ by

$0$ .

$0 + 0 - 6$

Step 2.1.2.2

Simplify by adding and subtracting.

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Step 2.1.2.2.1

Add

$0$ and

$0$ .

$0 - 6$

Step 2.1.2.2.2

Subtract

$6$ from

$0$ .

$- 6$

Step 2.2

Evaluate at

$x = 4$ .

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Step 2.2.1

Substitute

$4$ for

$x$ .

$\frac{2 {(4)}^{\frac{5}{2}}}{5} - \frac{2 {(4)}^{\frac{3}{2}}}{3} - 6$

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

Simplify the numerator.

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Step 2.2.2.1.1.1

Rewrite

$4$ as

$2^{2}$ .

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

Step 2.2.2.1.1.2

Multiply the exponents in

${(2^{2})}^{\frac{5}{2}}$ .

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Step 2.2.2.1.1.2.1

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

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

Step 2.2.2.1.1.2.2

Cancel the common factor of

$2$ .

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Step 2.2.2.1.1.2.2.1

Cancel the common factor.

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

Step 2.2.2.1.1.2.2.2

Rewrite the expression.

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

Step 2.2.2.1.1.3

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{2^{1 + 5}}{5} - \frac{2 {(4)}^{\frac{3}{2}}}{3} - 6$

Step 2.2.2.1.1.4

Add

$1$ and

$5$ .

$\frac{2^{6}}{5} - \frac{2 {(4)}^{\frac{3}{2}}}{3} - 6$

Step 2.2.2.1.2

Raise

$2$ to the power of

$6$ .

$\frac{64}{5} - \frac{2 {(4)}^{\frac{3}{2}}}{3} - 6$

Step 2.2.2.1.3

Simplify the numerator.

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Step 2.2.2.1.3.1

Rewrite

$4$ as

$2^{2}$ .

$\frac{64}{5} - \frac{2 \cdot {(2^{2})}^{\frac{3}{2}}}{3} - 6$

Step 2.2.2.1.3.2

Multiply the exponents in

${(2^{2})}^{\frac{3}{2}}$ .

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Step 2.2.2.1.3.2.1

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$\frac{64}{5} - \frac{2 \cdot 2^{2 (\frac{3}{2})}}{3} - 6$

Step 2.2.2.1.3.2.2

Cancel the common factor of

$2$ .

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Step 2.2.2.1.3.2.2.1

Cancel the common factor.

$\frac{64}{5} - \frac{2 \cdot 2^{2 (\frac{3}{2})}}{3} - 6$

Step 2.2.2.1.3.2.2.2

Rewrite the expression.

$\frac{64}{5} - \frac{2 \cdot 2^{3}}{3} - 6$

Step 2.2.2.1.3.3

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{64}{5} - \frac{2^{1 + 3}}{3} - 6$

Step 2.2.2.1.3.4

Add

$1$ and

$3$ .

$\frac{64}{5} - \frac{2^{4}}{3} - 6$

Step 2.2.2.1.4

Raise

$2$ to the power of

$4$ .

$\frac{64}{5} - \frac{16}{3} - 6$

Step 2.2.2.2

Find the common denominator.

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Step 2.2.2.2.1

Multiply

$\frac{64}{5}$ by

$\frac{3}{3}$ .

$\frac{64}{5} \cdot \frac{3}{3} - \frac{16}{3} - 6$

Step 2.2.2.2.2

Multiply

$\frac{64}{5}$ by

$\frac{3}{3}$ .

$\frac{64 \cdot 3}{5 \cdot 3} - \frac{16}{3} - 6$

Step 2.2.2.2.3

Multiply

$\frac{16}{3}$ by

$\frac{5}{5}$ .

$\frac{64 \cdot 3}{5 \cdot 3} - (\frac{16}{3} \cdot \frac{5}{5}) - 6$

Step 2.2.2.2.4

Multiply

$\frac{16}{3}$ by

$\frac{5}{5}$ .

$\frac{64 \cdot 3}{5 \cdot 3} - \frac{16 \cdot 5}{3 \cdot 5} - 6$

Step 2.2.2.2.5

Write

$- 6$ as a fraction with denominator

$1$ .

$\frac{64 \cdot 3}{5 \cdot 3} - \frac{16 \cdot 5}{3 \cdot 5} + \frac{- 6}{1}$

Step 2.2.2.2.6

Multiply

$\frac{- 6}{1}$ by

$\frac{15}{15}$ .

$\frac{64 \cdot 3}{5 \cdot 3} - \frac{16 \cdot 5}{3 \cdot 5} + \frac{- 6}{1} \cdot \frac{15}{15}$

Step 2.2.2.2.7

Multiply

$\frac{- 6}{1}$ by

$\frac{15}{15}$ .

$\frac{64 \cdot 3}{5 \cdot 3} - \frac{16 \cdot 5}{3 \cdot 5} + \frac{- 6 \cdot 15}{15}$

Step 2.2.2.2.8

Reorder the factors of

$5 \cdot 3$ .

$\frac{64 \cdot 3}{3 \cdot 5} - \frac{16 \cdot 5}{3 \cdot 5} + \frac{- 6 \cdot 15}{15}$

Step 2.2.2.2.9

Multiply

$3$ by

$5$ .

$\frac{64 \cdot 3}{15} - \frac{16 \cdot 5}{3 \cdot 5} + \frac{- 6 \cdot 15}{15}$

Step 2.2.2.2.10

Multiply

$3$ by

$5$ .

$\frac{64 \cdot 3}{15} - \frac{16 \cdot 5}{15} + \frac{- 6 \cdot 15}{15}$

Step 2.2.2.3

Combine the numerators over the common denominator.

$\frac{64 \cdot 3 - 16 \cdot 5 - 6 \cdot 15}{15}$

Step 2.2.2.4

Simplify each term.

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Step 2.2.2.4.1

Multiply

$64$ by

$3$ .

$\frac{192 - 16 \cdot 5 - 6 \cdot 15}{15}$

Step 2.2.2.4.2

Multiply

$- 16$ by

$5$ .

$\frac{192 - 80 - 6 \cdot 15}{15}$

Step 2.2.2.4.3

Multiply

$- 6$ by

$15$ .

$\frac{192 - 80 - 90}{15}$

Step 2.2.2.5

Simplify by subtracting numbers.

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Step 2.2.2.5.1

Subtract

$80$ from

$192$ .

$\frac{112 - 90}{15}$

Step 2.2.2.5.2

Subtract

$90$ from

$112$ .

$\frac{22}{15}$

Step 2.3

List all of the points.

$(0, - 6), (4, \frac{22}{15})$

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:

$(4, \frac{22}{15})$

Absolute Minimum:

$(1, - \frac{94}{15})$

Step 4