Calculus Examples

$f (x) = x^{\frac{2}{3}}$ ; $[- 1, 1]$

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 $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{2}{3}$ .

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

Step 1.1.1.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 1.1.1.3

Combine $- 1$ and $\frac{3}{3}$ .

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

Step 1.1.1.4

Combine the numerators over the common denominator.

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

Step 1.1.1.5

Simplify the numerator.

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Step 1.1.1.5.1

Multiply $- 1$ by $3$ .

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

Step 1.1.1.5.2

Subtract $3$ from $2$ .

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

Step 1.1.1.6

Move the negative in front of the fraction.

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

Step 1.1.1.7

Simplify.

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Step 1.1.1.7.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 1.1.1.7.2

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

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

Step 1.1.2

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

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

Step 1.2

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

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Step 1.2.1

Set the first derivative equal to $0$ .

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

Step 1.2.2

Set the numerator equal to zero.

$2 = 0$

Step 1.2.3

Since $2 \neq 0$ , 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 $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

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

Step 1.3.1.2

Anything raised to $1$ is the base itself.

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

Step 1.3.2

Set the denominator in $\frac{2}{3 \sqrt[3]{x}}$ equal to $0$ to find where the expression is undefined.

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

Step 1.3.3

Solve for $x$ .

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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.

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

Step 1.3.3.2

Simplify each side of the equation.

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Step 1.3.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x}$ as $x^{\frac{1}{3}}$ .

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

Step 1.3.3.2.2

Simplify the left side.

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Step 1.3.3.2.2.1

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

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Step 1.3.3.2.2.1.1

Apply the product rule to $3 x^{\frac{1}{3}}$ .

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

Step 1.3.3.2.2.1.2

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

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

Step 1.3.3.2.2.1.3

Multiply the exponents in ${(x^{\frac{1}{3}})}^{3}$ .

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Step 1.3.3.2.2.1.3.1

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

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

Step 1.3.3.2.2.1.3.2

Cancel the common factor of $3$ .

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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 $x = - 1$ .

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Step 2.1.1

Substitute $- 1$ for $x$ .

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

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 $- 1$ as ${(- 1)}^{3}$ .

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

Step 2.1.2.1.2

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

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

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.

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

Step 2.1.2.2.2

Rewrite the expression.

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

Step 2.1.2.3

Raise $- 1$ to the power of $2$ .

$1$

Step 2.2

Evaluate at $x = 1$ .

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Step 2.2.1

Substitute $1$ for $x$ .

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

Step 2.2.2

One to any power is one.

$1$

Step 2.3

List all of the points.

$(- 1, 1), (1, 1)$

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: $(- 1, 1), (1, 1)$

Absolute Minimum: $(0, 0)$

Step 4