Calculus Examples

y = 4 \sqrt[3]{x - 1} - 3

$y = 4 \sqrt[3]{x - 1} - 3$ ;

$[- 10, 10]$

Step 1

Find the critical points.

Tap for more steps...

Step 1.1

Find the first derivative.

Tap for more steps...

Step 1.1.1

Find the first derivative.

Tap for more steps...

Step 1.1.1.1

By the Sum Rule, the derivative of

$4 \sqrt[3]{x - 1} - 3$ with respect to

$x$ is

$\frac{d}{d x} [4 \sqrt[3]{x - 1}] + \frac{d}{d x} [- 3]$ .

$\frac{d}{d x} [4 \sqrt[3]{x - 1}] + \frac{d}{d x} [- 3]$

Step 1.1.1.2

Evaluate

$\frac{d}{d x} [4 \sqrt[3]{x - 1}]$ .

Tap for more steps...

Step 1.1.1.2.1

Use

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

$\sqrt[3]{x - 1}$ as

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

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

Step 1.1.1.2.2

Since

$4$ is constant with respect to

$x$ , the derivative of

$4 {(x - 1)}^{\frac{1}{3}}$ with respect to

$x$ is

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

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

Step 1.1.1.2.3

Differentiate using the chain rule, which states that

$\frac{d}{d x} [f (g (x))]$ is

$f' (g (x)) g' (x)$ where

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

$g (x) = x - 1$ .

Tap for more steps...

Step 1.1.1.2.3.1

To apply the Chain Rule, set

$u$ as

$x - 1$ .

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

Step 1.1.1.2.3.2

Differentiate using the Power Rule which states that

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

$n u^{n - 1}$ where

$n = \frac{1}{3}$ .

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

Step 1.1.1.2.3.3

Replace all occurrences of

$u$ with

$x - 1$ .

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

Step 1.1.1.2.4

By the Sum Rule, the derivative of

$x - 1$ with respect to

$x$ is

$\frac{d}{d x} [x] + \frac{d}{d x} [- 1]$ .

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

Step 1.1.1.2.5

Differentiate using the Power Rule which states that

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

$n x^{n - 1}$ where

$n = 1$ .

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

Step 1.1.1.2.6

Since

$- 1$ is constant with respect to

$x$ , the derivative of

$- 1$ with respect to

$x$ is

$0$ .

$4 (\frac{1}{3} {(x - 1)}^{\frac{1}{3} - 1} (1 + 0)) + \frac{d}{d x} [- 3]$

Step 1.1.1.2.7

To write

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

$\frac{3}{3}$ .

$4 (\frac{1}{3} {(x - 1)}^{\frac{1}{3} - 1 \cdot \frac{3}{3}} (1 + 0)) + \frac{d}{d x} [- 3]$

Step 1.1.1.2.8

Combine

$- 1$ and

$\frac{3}{3}$ .

$4 (\frac{1}{3} {(x - 1)}^{\frac{1}{3} + \frac{- 1 \cdot 3}{3}} (1 + 0)) + \frac{d}{d x} [- 3]$

Step 1.1.1.2.9

Combine the numerators over the common denominator.

$4 (\frac{1}{3} {(x - 1)}^{\frac{1 - 1 \cdot 3}{3}} (1 + 0)) + \frac{d}{d x} [- 3]$

Step 1.1.1.2.10

Simplify the numerator.

Tap for more steps...

Step 1.1.1.2.10.1

Multiply

$- 1$ by

$3$ .

$4 (\frac{1}{3} {(x - 1)}^{\frac{1 - 3}{3}} (1 + 0)) + \frac{d}{d x} [- 3]$

Step 1.1.1.2.10.2

Subtract

$3$ from

$1$ .

$4 (\frac{1}{3} {(x - 1)}^{\frac{- 2}{3}} (1 + 0)) + \frac{d}{d x} [- 3]$

Step 1.1.1.2.11

Move the negative in front of the fraction.

$4 (\frac{1}{3} {(x - 1)}^{- \frac{2}{3}} (1 + 0)) + \frac{d}{d x} [- 3]$

Step 1.1.1.2.12

Add

$1$ and

$0$ .

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

Step 1.1.1.2.13

Combine

$\frac{1}{3}$ and

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

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

Step 1.1.1.2.14

Multiply

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

$1$ .

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

Step 1.1.1.2.15

Move

${(x - 1)}^{- \frac{2}{3}}$ to the denominator using the negative exponent rule

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

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

Step 1.1.1.2.16

Combine

$4$ and

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

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

Step 1.1.1.3

Differentiate using the Constant Rule.

Tap for more steps...

Step 1.1.1.3.1

Since

$- 3$ is constant with respect to

$x$ , the derivative of

$- 3$ with respect to

$x$ is

$0$ .

$\frac{4}{3 {(x - 1)}^{\frac{2}{3}}} + 0$

Step 1.1.1.3.2

Add

$\frac{4}{3 {(x - 1)}^{\frac{2}{3}}}$ and

$0$ .

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

Step 1.1.2

The first derivative of

$f (x)$ with respect to

$x$ is

$\frac{4}{3 {(x - 1)}^{\frac{2}{3}}}$ .

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

Step 1.2

Set the first derivative equal to

$0$ then solve the equation

$\frac{4}{3 {(x - 1)}^{\frac{2}{3}}} = 0$ .

Tap for more steps...

Step 1.2.1

Set the first derivative equal to

$0$ .

$\frac{4}{3 {(x - 1)}^{\frac{2}{3}}} = 0$

Step 1.2.2

Set the numerator equal to zero.

$4 = 0$

Step 1.2.3

Since

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

No solution

Step 1.3

Find the values where the derivative is undefined.

Tap for more steps...

Step 1.3.1

Apply the rule

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

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

Step 1.3.2

Set the denominator in

$\frac{4}{3 \sqrt[3]{{(x - 1)}^{2}}}$ equal to

$0$ to find where the expression is undefined.

$3 \sqrt[3]{{(x - 1)}^{2}} = 0$

Step 1.3.3

Solve for

$x$ .

Tap for more steps...

Step 1.3.3.1

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

${(3 \sqrt[3]{{(x - 1)}^{2}})}^{3} = 0^{3}$

Step 1.3.3.2

Simplify each side of the equation.

Tap for more steps...

Step 1.3.3.2.1

Use

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

$\sqrt[3]{{(x - 1)}^{2}}$ as

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

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

Step 1.3.3.2.2

Simplify the left side.

Tap for more steps...

Step 1.3.3.2.2.1

Simplify

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

Tap for more steps...

Step 1.3.3.2.2.1.1

Apply the product rule to

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

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

Step 1.3.3.2.2.1.2

Raise

$3$ to the power of

$3$ .

$27 {({(x - 1)}^{\frac{2}{3}})}^{3} = 0^{3}$

Step 1.3.3.2.2.1.3

Multiply the exponents in

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

Tap for more steps...

Step 1.3.3.2.2.1.3.1

Apply the power rule and multiply exponents,

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

$27 {(x - 1)}^{\frac{2}{3} \cdot 3} = 0^{3}$

Step 1.3.3.2.2.1.3.2

Cancel the common factor of

$3$ .

Tap for more steps...

Step 1.3.3.2.2.1.3.2.1

Cancel the common factor.

$27 {(x - 1)}^{\frac{2}{3} \cdot 3} = 0^{3}$

Step 1.3.3.2.2.1.3.2.2

Rewrite the expression.

$27 {(x - 1)}^{2} = 0^{3}$

Step 1.3.3.2.3

Simplify the right side.

Tap for more steps...

Step 1.3.3.2.3.1

Raising

$0$ to any positive power yields

$0$ .

$27 {(x - 1)}^{2} = 0$

Step 1.3.3.3

Solve for

$x$ .

Tap for more steps...

Step 1.3.3.3.1

Divide each term in

$27 {(x - 1)}^{2} = 0$ by

$27$ and simplify.

Tap for more steps...

Step 1.3.3.3.1.1

Divide each term in

$27 {(x - 1)}^{2} = 0$ by

$27$ .

$\frac{27 {(x - 1)}^{2}}{27} = \frac{0}{27}$

Step 1.3.3.3.1.2

Simplify the left side.

Tap for more steps...

Step 1.3.3.3.1.2.1

Cancel the common factor of

$27$ .

Tap for more steps...

Step 1.3.3.3.1.2.1.1

Cancel the common factor.

$\frac{27 {(x - 1)}^{2}}{27} = \frac{0}{27}$

Step 1.3.3.3.1.2.1.2

Divide

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

$1$ .

${(x - 1)}^{2} = \frac{0}{27}$

Step 1.3.3.3.1.3

Simplify the right side.

Tap for more steps...

Step 1.3.3.3.1.3.1

Divide

$0$ by

$27$ .

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

Step 1.3.3.3.2

Set the

$x - 1$ equal to

$0$ .

$x - 1 = 0$

Step 1.3.3.3.3

Add

$1$ to both sides of the equation.

$x = 1$

Step 1.4

Evaluate

$4 \sqrt[3]{x - 1} - 3$ at each

$x$ value where the derivative is

$0$ or undefined.

Tap for more steps...

Step 1.4.1

Evaluate at

$x = 1$ .

Tap for more steps...

Step 1.4.1.1

Substitute

$1$ for

$x$ .

$4 \sqrt[3]{(1) - 1} - 3$

Step 1.4.1.2

Simplify.

Tap for more steps...

Step 1.4.1.2.1

Simplify each term.

Tap for more steps...

Step 1.4.1.2.1.1

Subtract

$1$ from

$1$ .

$4 \sqrt[3]{0} - 3$

Step 1.4.1.2.1.2

Rewrite

$0$ as

$0^{3}$ .

$4 \sqrt[3]{0^{3}} - 3$

Step 1.4.1.2.1.3

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

$4 \cdot 0 - 3$

Step 1.4.1.2.1.4

Multiply

$4$ by

$0$ .

$0 - 3$

Step 1.4.1.2.2

Subtract

$3$ from

$0$ .

$- 3$

Step 1.4.2

List all of the points.

$(1, - 3)$

Step 2

Evaluate at the included endpoints.

Tap for more steps...

Step 2.1

Evaluate at

$x = - 10$ .

Tap for more steps...

Step 2.1.1

Substitute

$- 10$ for

$x$ .

$4 \sqrt[3]{(- 10) - 1} - 3$

Step 2.1.2

Simplify each term.

Tap for more steps...

Step 2.1.2.1

Subtract

$1$ from

$- 10$ .

$4 \sqrt[3]{- 11} - 3$

Step 2.1.2.2

Rewrite

$- 11$ as

${(- 1)}^{3} \cdot 11$ .

Tap for more steps...

Step 2.1.2.2.1

Rewrite

$- 11$ as

$- 1 (11)$ .

$4 \sqrt[3]{- 1 (11)} - 3$

Step 2.1.2.2.2

Rewrite

$- 1$ as

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

$4 \sqrt[3]{{(- 1)}^{3} \cdot 11} - 3$

Step 2.1.2.3

Pull terms out from under the radical.

$4 (- 1 \sqrt[3]{11}) - 3$

Step 2.1.2.4

Rewrite

$- 1 \sqrt[3]{11}$ as

$- \sqrt[3]{11}$ .

$4 (- \sqrt[3]{11}) - 3$

Step 2.1.2.5

Multiply

$- 1$ by

$4$ .

$- 4 \sqrt[3]{11} - 3$

Step 2.2

Evaluate at

$x = 10$ .

Tap for more steps...

Step 2.2.1

Substitute

$10$ for

$x$ .

$4 \sqrt[3]{(10) - 1} - 3$

Step 2.2.2

Subtract

$1$ from

$10$ .

$4 \sqrt[3]{9} - 3$

Step 2.3

List all of the points.

$(- 10, - 4 \sqrt[3]{11} - 3), (10, 4 \sqrt[3]{9} - 3)$

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:

$(10, 4 \sqrt[3]{9} - 3)$

Absolute Minimum:

$(- 10, - 4 \sqrt[3]{11} - 3)$

Step 4