Calculus Examples

$f (x) = 12 \sqrt[3]{x}$

Step 1

Find the first derivative of the function.

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Step 1.1

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

$\frac{d}{d x} [12 x^{\frac{1}{3}}]$

Step 1.2

Since $12$ is constant with respect to $x$ , the derivative of $12 x^{\frac{1}{3}}$ with respect to $x$ is $12 \frac{d}{d x} [x^{\frac{1}{3}}]$ .

$12 \frac{d}{d x} [x^{\frac{1}{3}}]$

Step 1.3

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

$12 (\frac{1}{3} x^{\frac{1}{3} - 1})$

Step 1.4

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

$12 (\frac{1}{3} x^{\frac{1}{3} - 1 \cdot \frac{3}{3}})$

Step 1.5

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

$12 (\frac{1}{3} x^{\frac{1}{3} + \frac{- 1 \cdot 3}{3}})$

Step 1.6

Combine the numerators over the common denominator.

$12 (\frac{1}{3} x^{\frac{1 - 1 \cdot 3}{3}})$

Step 1.7

Simplify the numerator.

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Step 1.7.1

Multiply $- 1$ by $3$ .

$12 (\frac{1}{3} x^{\frac{1 - 3}{3}})$

Step 1.7.2

Subtract $3$ from $1$ .

$12 (\frac{1}{3} x^{\frac{- 2}{3}})$

Step 1.8

Move the negative in front of the fraction.

$12 (\frac{1}{3} x^{- \frac{2}{3}})$

Step 1.9

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

$12 \frac{x^{- \frac{2}{3}}}{3}$

Step 1.10

Combine $12$ and $\frac{x^{- \frac{2}{3}}}{3}$ .

$\frac{12 x^{- \frac{2}{3}}}{3}$

Step 1.11

Move $x^{- \frac{2}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{12}{3 x^{\frac{2}{3}}}$

Step 1.12

Factor $3$ out of $12$ .

$\frac{3 \cdot 4}{3 x^{\frac{2}{3}}}$

Step 1.13

Cancel the common factors.

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Step 1.13.1

Factor $3$ out of $3 x^{\frac{2}{3}}$ .

$\frac{3 \cdot 4}{3 (x^{\frac{2}{3}})}$

Step 1.13.2

Cancel the common factor.

$\frac{3 \cdot 4}{3 x^{\frac{2}{3}}}$

Step 1.13.3

Rewrite the expression.

$\frac{4}{x^{\frac{2}{3}}}$

Step 2

Find the second derivative of the function.

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Step 2.1

Since $4$ is constant with respect to $x$ , the derivative of $\frac{4}{x^{\frac{2}{3}}}$ with respect to $x$ is $4 \frac{d}{d x} [\frac{1}{x^{\frac{2}{3}}}]$ .

$f'' (x) = 4 \frac{d}{d x} (\frac{1}{x^{\frac{2}{3}}})$

Step 2.2

Apply basic rules of exponents.

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Step 2.2.1

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

$f'' (x) = 4 \frac{d}{d x} ({(x^{\frac{2}{3}})}^{- 1})$

Step 2.2.2

Multiply the exponents in ${(x^{\frac{2}{3}})}^{- 1}$ .

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Step 2.2.2.1

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

$f'' (x) = 4 \frac{d}{d x} (x^{\frac{2}{3} \cdot - 1})$

Step 2.2.2.2

Multiply $\frac{2}{3} \cdot - 1$ .

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Step 2.2.2.2.1

Combine $\frac{2}{3}$ and $- 1$ .

$f'' (x) = 4 \frac{d}{d x} (x^{\frac{2 \cdot - 1}{3}})$

Step 2.2.2.2.2

Multiply $2$ by $- 1$ .

$f'' (x) = 4 \frac{d}{d x} (x^{\frac{- 2}{3}})$

Step 2.2.2.3

Move the negative in front of the fraction.

$f'' (x) = 4 \frac{d}{d x} (x^{- \frac{2}{3}})$

Step 2.3

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

$f'' (x) = 4 (- \frac{2}{3} x^{- \frac{2}{3} - 1})$

Step 2.4

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

$f'' (x) = 4 (- \frac{2}{3} x^{- \frac{2}{3} - 1 \cdot \frac{3}{3}})$

Step 2.5

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

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

Step 2.6

Combine the numerators over the common denominator.

$f'' (x) = 4 (- \frac{2}{3} x^{\frac{- 2 - 1 \cdot 3}{3}})$

Step 2.7

Simplify the numerator.

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Step 2.7.1

Multiply $- 1$ by $3$ .

$f'' (x) = 4 (- \frac{2}{3} x^{\frac{- 2 - 3}{3}})$

Step 2.7.2

Subtract $3$ from $- 2$ .

$f'' (x) = 4 (- \frac{2}{3} x^{\frac{- 5}{3}})$

Step 2.8

Move the negative in front of the fraction.

$f'' (x) = 4 (- \frac{2}{3} x^{- \frac{5}{3}})$

Step 2.9

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

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

Step 2.10

Multiply $- 1$ by $4$ .

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

Step 2.11

Combine $- 4$ and $\frac{x^{- \frac{5}{3}} \cdot 2}{3}$ .

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

Step 2.12

Simplify the expression.

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Step 2.12.1

Multiply $2$ by $- 4$ .

$f'' (x) = \frac{- 8 x^{- \frac{5}{3}}}{3}$

Step 2.12.2

Move $x^{- \frac{5}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (x) = \frac{- 8}{3 x^{\frac{5}{3}}}$

Step 2.12.3

Move the negative in front of the fraction.

$f'' (x) = - \frac{8}{3 x^{\frac{5}{3}}}$

Step 3

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$\frac{4}{x^{\frac{2}{3}}} = 0$

Step 4

Find the first derivative.

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Step 4.1

Find the first derivative.

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Step 4.1.1

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

$\frac{d}{d x} [12 x^{\frac{1}{3}}]$

Step 4.1.2

Since $12$ is constant with respect to $x$ , the derivative of $12 x^{\frac{1}{3}}$ with respect to $x$ is $12 \frac{d}{d x} [x^{\frac{1}{3}}]$ .

$12 \frac{d}{d x} [x^{\frac{1}{3}}]$

Step 4.1.3

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

$12 (\frac{1}{3} x^{\frac{1}{3} - 1})$

Step 4.1.4

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

$12 (\frac{1}{3} x^{\frac{1}{3} - 1 \cdot \frac{3}{3}})$

Step 4.1.5

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

$12 (\frac{1}{3} x^{\frac{1}{3} + \frac{- 1 \cdot 3}{3}})$

Step 4.1.6

Combine the numerators over the common denominator.

$12 (\frac{1}{3} x^{\frac{1 - 1 \cdot 3}{3}})$

Step 4.1.7

Simplify the numerator.

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Step 4.1.7.1

Multiply $- 1$ by $3$ .

$12 (\frac{1}{3} x^{\frac{1 - 3}{3}})$

Step 4.1.7.2

Subtract $3$ from $1$ .

$12 (\frac{1}{3} x^{\frac{- 2}{3}})$

Step 4.1.8

Move the negative in front of the fraction.

$12 (\frac{1}{3} x^{- \frac{2}{3}})$

Step 4.1.9

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

$12 \frac{x^{- \frac{2}{3}}}{3}$

Step 4.1.10

Combine $12$ and $\frac{x^{- \frac{2}{3}}}{3}$ .

$\frac{12 x^{- \frac{2}{3}}}{3}$

Step 4.1.11

Move $x^{- \frac{2}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{12}{3 x^{\frac{2}{3}}}$

Step 4.1.12

Factor $3$ out of $12$ .

$\frac{3 \cdot 4}{3 x^{\frac{2}{3}}}$

Step 4.1.13

Cancel the common factors.

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Step 4.1.13.1

Factor $3$ out of $3 x^{\frac{2}{3}}$ .

$\frac{3 \cdot 4}{3 (x^{\frac{2}{3}})}$

Step 4.1.13.2

Cancel the common factor.

$\frac{3 \cdot 4}{3 x^{\frac{2}{3}}}$

Step 4.1.13.3

Rewrite the expression.

$f' (x) = \frac{4}{x^{\frac{2}{3}}}$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{4}{x^{\frac{2}{3}}}$ .

$\frac{4}{x^{\frac{2}{3}}}$

Step 5

Set the first derivative equal to $0$ then solve the equation $\frac{4}{x^{\frac{2}{3}}} = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$\frac{4}{x^{\frac{2}{3}}} = 0$

Step 5.2

Set the numerator equal to zero.

$4 = 0$

Step 5.3

Since $4 \neq 0$ , there are no solutions.

No solution

Step 6

Find the values where the derivative is undefined.

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Step 6.1

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

$\frac{4}{\sqrt[3]{x^{2}}}$

Step 6.2

Set the denominator in $\frac{4}{\sqrt[3]{x^{2}}}$ equal to $0$ to find where the expression is undefined.

$\sqrt[3]{x^{2}} = 0$

Step 6.3

Solve for $x$ .

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Step 6.3.1

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

${\sqrt[3]{x^{2}}}^{3} = 0^{3}$

Step 6.3.2

Simplify each side of the equation.

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Step 6.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{2}}$ as $x^{\frac{2}{3}}$ .

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

Step 6.3.2.2

Simplify the left side.

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Step 6.3.2.2.1

Multiply the exponents in ${(x^{\frac{2}{3}})}^{3}$ .

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Step 6.3.2.2.1.1

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

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

Step 6.3.2.2.1.2

Cancel the common factor of $3$ .

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Step 6.3.2.2.1.2.1

Cancel the common factor.

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

Step 6.3.2.2.1.2.2

Rewrite the expression.

$x^{2} = 0^{3}$

Step 6.3.2.3

Simplify the right side.

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Step 6.3.2.3.1

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

$x^{2} = 0$

Step 6.3.3

Solve for $x$ .

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Step 6.3.3.1

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

$x = \pm \sqrt{0}$

Step 6.3.3.2

Simplify $\pm \sqrt{0}$ .

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Step 6.3.3.2.1

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

$x = \pm \sqrt{0^{2}}$

Step 6.3.3.2.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Step 6.3.3.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 7

Critical points to evaluate.

$x = 0$

Step 8

Evaluate the second derivative at $x = 0$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- \frac{8}{3 {(0)}^{\frac{5}{3}}}$

Step 9

Evaluate the second derivative.

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Step 9.1

Simplify the expression.

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Step 9.1.1

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

$- \frac{8}{3 {(0^{3})}^{\frac{5}{3}}}$

Step 9.1.2

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

$- \frac{8}{3 \cdot 0^{3 (\frac{5}{3})}}$

Step 9.2

Cancel the common factor of $3$ .

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Step 9.2.1

Cancel the common factor.

$- \frac{8}{3 \cdot 0^{3 (\frac{5}{3})}}$

Step 9.2.2

Rewrite the expression.

$- \frac{8}{3 \cdot 0^{5}}$

Step 9.3

Simplify the expression.

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Step 9.3.1

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

$- \frac{8}{3 \cdot 0}$

Step 9.3.2

Multiply $3$ by $0$ .

$- \frac{8}{0}$

Step 9.3.3

The expression contains a division by $0$ . The expression is undefined.

Undefined

$- \frac{8}{0}$

Step 9.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

Step 10

Since there is at least one point with $0$ or undefined second derivative, apply the first derivative test.

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Step 10.1

Split $(- \infty, \infty)$ into separate intervals around the $x$ values that make the first derivative $0$ or undefined.

$(- \infty, 0) \cup (0, \infty)$

Step 10.2

Substitute any number, such as $- 2$ , from the interval $(- \infty, 0)$ in the first derivative $\frac{4}{x^{\frac{2}{3}}}$ to check if the result is negative or positive.

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Step 10.2.1

Replace the variable $x$ with $- 2$ in the expression.

$f' (- 2) = \frac{4}{{(- 2)}^{\frac{2}{3}}}$

Step 10.2.2

The final answer is $\frac{4}{{(- 2)}^{\frac{2}{3}}}$ .

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

Step 10.3

Substitute any number, such as $2$ , from the interval $(0, \infty)$ in the first derivative $\frac{4}{x^{\frac{2}{3}}}$ to check if the result is negative or positive.

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Step 10.3.1

Replace the variable $x$ with $2$ in the expression.

$f' (2) = \frac{4}{{(2)}^{\frac{2}{3}}}$

Step 10.3.2

Simplify the result.

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Step 10.3.2.1

Remove parentheses.

$f' (2) = \frac{4}{2^{\frac{2}{3}}}$

Step 10.3.2.2

The final answer is $\frac{4}{2^{\frac{2}{3}}}$ .

$\frac{4}{2^{\frac{2}{3}}}$

Step 10.4

Since the first derivative did not change signs around $x = 0$ , this is not a local maximum or minimum.

Not a local maximum or minimum

Step 10.5

No local maxima or minima found for $f (x) = 12 \sqrt[3]{x}$ .

No local maxima or minima

Step 11