Calculus Examples

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

$f (x) = x^{\frac{2}{3}}$ ;

[- 27, 27]

$[- 27, 27]$

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

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{2}{3}

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

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

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

Step 1.1.1.2

To write

- 1

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

\frac{3}{3}

$\frac{3}{3}$ .

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

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

Step 1.1.1.3

Combine

- 1

$- 1$ and

\frac{3}{3}

$\frac{3}{3}$ .

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

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

Step 1.1.1.4

Combine the numerators over the common denominator.

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

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

Step 1.1.1.5

Simplify the numerator.

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Step 1.1.1.5.1

Multiply

- 1

$- 1$ by

3

$3$ .

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

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

Step 1.1.1.5.2

Subtract

3

$3$ from

2

$2$ .

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

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

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

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

Step 1.1.1.6

Move the negative in front of the fraction.

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

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

Step 1.1.1.7

Simplify.

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Step 1.1.1.7.1

Rewrite the expression using the negative exponent rule

b^{- n} = \frac{1}{b^{n}}

$b^{- n} = \frac{1}{b^{n}}$ .

\frac{2}{3} \cdot \frac{1}{x^{\frac{1}{3}}}

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

Step 1.1.1.7.2

Multiply

\frac{2}{3}

$\frac{2}{3}$ by

\frac{1}{x^{\frac{1}{3}}}

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

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

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

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

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

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

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

Step 1.1.2

The first derivative of

f (x)

$f (x)$ with respect to

x

$x$ is

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

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

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

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

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

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

Step 1.2

Set the first derivative equal to

0

$0$ then solve the equation

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

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

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Step 1.2.1

Set the first derivative equal to

0

$0$ .

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

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

Step 1.2.2

Set the numerator equal to zero.

2 = 0

$2 = 0$

Step 1.2.3

Since

2 \neq 0

$2 \neq 0$ , there are no solutions.

No solution

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}}

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

\frac{2}{3 \sqrt[3]{x^{1}}}

$\frac{2}{3 \sqrt[3]{x^{1}}}$

Step 1.3.1.2

Anything raised to

1

$1$ is the base itself.

\frac{2}{3 \sqrt[3]{x}}

$\frac{2}{3 \sqrt[3]{x}}$

\frac{2}{3 \sqrt[3]{x}}

$\frac{2}{3 \sqrt[3]{x}}$

Step 1.3.2

Set the denominator in

\frac{2}{3 \sqrt[3]{x}}

$\frac{2}{3 \sqrt[3]{x}}$ equal to

0

$0$ to find where the expression is undefined.

3 \sqrt[3]{x} = 0

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

Step 1.3.3

Solve for

x

$x$ .

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Step 1.3.3.1

To remove the radical on the left side of the equation, cube both sides of the equation.

{(3 \sqrt[3]{x})}^{3} = 0^{3}

${(3 \sqrt[3]{x})}^{3} = 0^{3}$

Step 1.3.3.2

Simplify each side of the equation.

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Step 1.3.3.2.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt[3]{x}

$\sqrt[3]{x}$ as

x^{\frac{1}{3}}

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

{(3 x^{\frac{1}{3}})}^{3} = 0^{3}

${(3 x^{\frac{1}{3}})}^{3} = 0^{3}$

Step 1.3.3.2.2

Simplify the left side.

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Step 1.3.3.2.2.1

Simplify

{(3 x^{\frac{1}{3}})}^{3}

${(3 x^{\frac{1}{3}})}^{3}$ .

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Step 1.3.3.2.2.1.1

Apply the product rule to

3 x^{\frac{1}{3}}

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

3^{3} {(x^{\frac{1}{3}})}^{3} = 0^{3}

$3^{3} {(x^{\frac{1}{3}})}^{3} = 0^{3}$

Step 1.3.3.2.2.1.2

Raise

3

$3$ to the power of

3

$3$ .

27 {(x^{\frac{1}{3}})}^{3} = 0^{3}

$27 {(x^{\frac{1}{3}})}^{3} = 0^{3}$

Step 1.3.3.2.2.1.3

Multiply the exponents in

{(x^{\frac{1}{3}})}^{3}

${(x^{\frac{1}{3}})}^{3}$ .

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Step 1.3.3.2.2.1.3.1

Apply the power rule and multiply exponents,

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

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

27 x^{\frac{1}{3} \cdot 3} = 0^{3}

$27 x^{\frac{1}{3} \cdot 3} = 0^{3}$

Step 1.3.3.2.2.1.3.2

Cancel the common factor of

3

$3$ .

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Step 1.3.3.2.2.1.3.2.1

Cancel the common factor.

$27 x^{\frac{1}{3} \cdot 3} = 0^{3}$

Step 1.3.3.2.2.1.3.2.2

Rewrite the expression.

$27 x^{1} = 0^{3}$

Step 1.3.3.2.2.1.4

Simplify.

$27 x = 0^{3}$

Step 1.3.3.2.3

Simplify the right side.

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Step 1.3.3.2.3.1

Raising

$0$ to any positive power yields

$0$ .

$27 x = 0$

Step 1.3.3.3

Divide each term in

$27 x = 0$ by

$27$ and simplify.

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Step 1.3.3.3.1

Divide each term in

$27 x = 0$ by

$27$ .

$\frac{27 x}{27} = \frac{0}{27}$

Step 1.3.3.3.2

Simplify the left side.

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Step 1.3.3.3.2.1

Cancel the common factor of

$27$ .

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Step 1.3.3.3.2.1.1

Cancel the common factor.

$\frac{27 x}{27} = \frac{0}{27}$

Step 1.3.3.3.2.1.2

Divide

$x$ by

$1$ .

$x = \frac{0}{27}$

Step 1.3.3.3.3

Simplify the right side.

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Step 1.3.3.3.3.1

Divide

$0$ by

$27$ .

$x = 0$

Step 1.4

Evaluate

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

$x$ .

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

Step 1.4.1.2

Simplify.

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Step 1.4.1.2.1

Simplify the expression.

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Step 1.4.1.2.1.1

Rewrite

$0$ as

$0^{3}$ .

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

Step 1.4.1.2.1.2

Apply the power rule and multiply exponents,

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

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

Step 1.4.1.2.2

Cancel the common factor of

$3$ .

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Step 1.4.1.2.2.1

Cancel the common factor.

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

Step 1.4.1.2.2.2

Rewrite the expression.

$0^{2}$

Step 1.4.1.2.3

Raising

$0$ to any positive power yields

$0$ .

$0$

Step 1.4.2

List all of the points.

$(0, 0)$

Step 2

Evaluate at the included endpoints.

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Step 2.1

Evaluate at

$x = - 27$ .

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Step 2.1.1

Substitute

$- 27$ for

$x$ .

${(- 27)}^{\frac{2}{3}}$

Step 2.1.2

Simplify.

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Step 2.1.2.1

Simplify the expression.

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Step 2.1.2.1.1

Rewrite

$- 27$ as

${(- 3)}^{3}$ .

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

Step 2.1.2.1.2

Apply the power rule and multiply exponents,

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

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

Step 2.1.2.2

Cancel the common factor of

$3$ .

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Step 2.1.2.2.1

Cancel the common factor.

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

Step 2.1.2.2.2

Rewrite the expression.

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

Step 2.1.2.3

Raise

$- 3$ to the power of

$2$ .

$9$

Step 2.2

Evaluate at

$x = 27$ .

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Step 2.2.1

Substitute

$27$ for

$x$ .

${(27)}^{\frac{2}{3}}$

Step 2.2.2

Simplify.

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Step 2.2.2.1

Simplify the expression.

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Step 2.2.2.1.1

Rewrite

$27$ as

$3^{3}$ .

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

Step 2.2.2.1.2

Apply the power rule and multiply exponents,

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

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

Step 2.2.2.2

Cancel the common factor of

$3$ .

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Step 2.2.2.2.1

Cancel the common factor.

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

Step 2.2.2.2.2

Rewrite the expression.

$3^{2}$

Step 2.2.2.3

Raise

$3$ to the power of

$2$ .

$9$

Step 2.3

List all of the points.

$(- 27, 9), (27, 9)$

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:

$(- 27, 9), (27, 9)$

Absolute Minimum:

$(0, 0)$

Step 4