Calculus Examples

f (x) = 4 x^{3} - x^{4} + x + 5

$f (x) = 4 x^{3} - x^{4} + x + 5$

Step 1

Find the first derivative.

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Step 1.1

By the Sum Rule, the derivative of

4 x^{3} - x^{4} + x + 5

$4 x^{3} - x^{4} + x + 5$ with respect to

x

$x$ is

\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- x^{4}] + \frac{d}{d x} [x] + \frac{d}{d x} [5]

$\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- x^{4}] + \frac{d}{d x} [x] + \frac{d}{d x} [5]$ .

\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- x^{4}] + \frac{d}{d x} [x] + \frac{d}{d x} [5]

$\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- x^{4}] + \frac{d}{d x} [x] + \frac{d}{d x} [5]$

Step 1.2

Evaluate

\frac{d}{d x} [4 x^{3}]

$\frac{d}{d x} [4 x^{3}]$ .

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Step 1.2.1

Since

4

$4$ is constant with respect to

x

$x$ , the derivative of

4 x^{3}

$4 x^{3}$ with respect to

x

$x$ is

4 \frac{d}{d x} [x^{3}]

$4 \frac{d}{d x} [x^{3}]$ .

4 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- x^{4}] + \frac{d}{d x} [x] + \frac{d}{d x} [5]

$4 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- x^{4}] + \frac{d}{d x} [x] + \frac{d}{d x} [5]$

Step 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 = 3

$n = 3$ .

4 (3 x^{2}) + \frac{d}{d x} [- x^{4}] + \frac{d}{d x} [x] + \frac{d}{d x} [5]

$4 (3 x^{2}) + \frac{d}{d x} [- x^{4}] + \frac{d}{d x} [x] + \frac{d}{d x} [5]$

Step 1.2.3

Multiply

3

$3$ by

4

$4$ .

12 x^{2} + \frac{d}{d x} [- x^{4}] + \frac{d}{d x} [x] + \frac{d}{d x} [5]

$12 x^{2} + \frac{d}{d x} [- x^{4}] + \frac{d}{d x} [x] + \frac{d}{d x} [5]$

12 x^{2} + \frac{d}{d x} [- x^{4}] + \frac{d}{d x} [x] + \frac{d}{d x} [5]

$12 x^{2} + \frac{d}{d x} [- x^{4}] + \frac{d}{d x} [x] + \frac{d}{d x} [5]$

Step 1.3

Evaluate

\frac{d}{d x} [- x^{4}]

$\frac{d}{d x} [- x^{4}]$ .

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Step 1.3.1

Since

- 1

$- 1$ is constant with respect to

x

$x$ , the derivative of

- x^{4}

$- x^{4}$ with respect to

x

$x$ is

- \frac{d}{d x} [x^{4}]

$- \frac{d}{d x} [x^{4}]$ .

12 x^{2} - \frac{d}{d x} [x^{4}] + \frac{d}{d x} [x] + \frac{d}{d x} [5]

$12 x^{2} - \frac{d}{d x} [x^{4}] + \frac{d}{d x} [x] + \frac{d}{d x} [5]$

Step 1.3.2

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 = 4

$n = 4$ .

12 x^{2} - (4 x^{3}) + \frac{d}{d x} [x] + \frac{d}{d x} [5]

$12 x^{2} - (4 x^{3}) + \frac{d}{d x} [x] + \frac{d}{d x} [5]$

Step 1.3.3

Multiply

4

$4$ by

- 1

$- 1$ .

12 x^{2} - 4 x^{3} + \frac{d}{d x} [x] + \frac{d}{d x} [5]

$12 x^{2} - 4 x^{3} + \frac{d}{d x} [x] + \frac{d}{d x} [5]$

12 x^{2} - 4 x^{3} + \frac{d}{d x} [x] + \frac{d}{d x} [5]

$12 x^{2} - 4 x^{3} + \frac{d}{d x} [x] + \frac{d}{d x} [5]$

Step 1.4

Differentiate.

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Step 1.4.1

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 = 1

$n = 1$ .

12 x^{2} - 4 x^{3} + 1 + \frac{d}{d x} [5]

$12 x^{2} - 4 x^{3} + 1 + \frac{d}{d x} [5]$

Step 1.4.2

Since

5

$5$ is constant with respect to

x

$x$ , the derivative of

5

$5$ with respect to

x

$x$ is

0

$0$ .

12 x^{2} - 4 x^{3} + 1 + 0

$12 x^{2} - 4 x^{3} + 1 + 0$

12 x^{2} - 4 x^{3} + 1 + 0

$12 x^{2} - 4 x^{3} + 1 + 0$

Step 1.5

Simplify.

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Step 1.5.1

Add

12 x^{2} - 4 x^{3} + 1

$12 x^{2} - 4 x^{3} + 1$ and

0

$0$ .

12 x^{2} - 4 x^{3} + 1

$12 x^{2} - 4 x^{3} + 1$

Step 1.5.2

Reorder terms.

- 4 x^{3} + 12 x^{2} + 1

$- 4 x^{3} + 12 x^{2} + 1$

- 4 x^{3} + 12 x^{2} + 1

$- 4 x^{3} + 12 x^{2} + 1$

- 4 x^{3} + 12 x^{2} + 1

$- 4 x^{3} + 12 x^{2} + 1$

Step 2

Graph each side of the equation. The solution is the x-value of the point of intersection.

x \approx 3.02727941

$x \approx 3.02727941$

Step 3

Split

(- \infty, \infty)

$(- \infty, \infty)$ into separate intervals around the

x

$x$ values that make the first derivative

0

$0$ or undefined.

(- \infty, 3.02727941) \cup (3.02727941, \infty)

$(- \infty, 3.02727941) \cup (3.02727941, \infty)$

Step 4

Substitute any number, such as

0

$0$ , from the interval

(- \infty, 3.02727941)

$(- \infty, 3.02727941)$ in the first derivative

- 4 x^{3} + 12 x^{2} + 1

$- 4 x^{3} + 12 x^{2} + 1$ to check if the result is negative or positive.

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Step 4.1

Replace the variable

x

$x$ with

0

$0$ in the expression.

f' (0) = - 4 {(0)}^{3} + 12 {(0)}^{2} + 1

$f' (0) = - 4 {(0)}^{3} + 12 {(0)}^{2} + 1$

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Raising

0

$0$ to any positive power yields

0

$0$ .

f' (0) = - 4 \cdot 0 + 12 {(0)}^{2} + 1

$f' (0) = - 4 \cdot 0 + 12 {(0)}^{2} + 1$

Step 4.2.1.2

Multiply

- 4

$- 4$ by

0

$0$ .

f' (0) = 0 + 12 {(0)}^{2} + 1

$f' (0) = 0 + 12 {(0)}^{2} + 1$

Step 4.2.1.3

Raising

0

$0$ to any positive power yields

0

$0$ .

f' (0) = 0 + 12 \cdot 0 + 1

$f' (0) = 0 + 12 \cdot 0 + 1$

Step 4.2.1.4

Multiply

12

$12$ by

0

$0$ .

f' (0) = 0 + 0 + 1

$f' (0) = 0 + 0 + 1$

f' (0) = 0 + 0 + 1

$f' (0) = 0 + 0 + 1$

Step 4.2.2

Simplify by adding numbers.

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Step 4.2.2.1

Add

0

$0$ and

0

$0$ .

f' (0) = 0 + 1

$f' (0) = 0 + 1$

Step 4.2.2.2

Add

0

$0$ and

1

$1$ .

f' (0) = 1

$f' (0) = 1$

f' (0) = 1

$f' (0) = 1$

Step 4.2.3

The final answer is

1

$1$ .

1

$1$

1

$1$

1

$1$

Step 5

Substitute any number, such as

6

$6$ , from the interval

(3.02727941, \infty)

$(3.02727941, \infty)$ in the first derivative

- 4 x^{3} + 12 x^{2} + 1

$- 4 x^{3} + 12 x^{2} + 1$ to check if the result is negative or positive.

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Step 5.1

Replace the variable

x

$x$ with

6

$6$ in the expression.

f' (6) = - 4 {(6)}^{3} + 12 {(6)}^{2} + 1

$f' (6) = - 4 {(6)}^{3} + 12 {(6)}^{2} + 1$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Raise

6

$6$ to the power of

3

$3$ .

f' (6) = - 4 \cdot 216 + 12 {(6)}^{2} + 1

$f' (6) = - 4 \cdot 216 + 12 {(6)}^{2} + 1$

Step 5.2.1.2

Multiply

- 4

$- 4$ by

216

$216$ .

f' (6) = - 864 + 12 {(6)}^{2} + 1

$f' (6) = - 864 + 12 {(6)}^{2} + 1$

Step 5.2.1.3

Raise

6

$6$ to the power of

2

$2$ .

f' (6) = - 864 + 12 \cdot 36 + 1

$f' (6) = - 864 + 12 \cdot 36 + 1$

Step 5.2.1.4

Multiply

12

$12$ by

36

$36$ .

f' (6) = - 864 + 432 + 1

$f' (6) = - 864 + 432 + 1$

f' (6) = - 864 + 432 + 1

$f' (6) = - 864 + 432 + 1$

Step 5.2.2

Simplify by adding numbers.

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Step 5.2.2.1

Add

- 864

$- 864$ and

432

$432$ .

f' (6) = - 432 + 1

$f' (6) = - 432 + 1$

Step 5.2.2.2

Add

- 432

$- 432$ and

1

$1$ .

f' (6) = - 431

$f' (6) = - 431$

f' (6) = - 431

$f' (6) = - 431$

Step 5.2.3

The final answer is

- 431

$- 431$ .

- 431

$- 431$

- 431

$- 431$

- 431

$- 431$

Step 6

Since the first derivative changed signs from positive to negative around

x = 3.02727941

$x = 3.02727941$ , then there is a turning point at

x = 3.02727941

$x = 3.02727941$ .

Step 7

Find the y-coordinate of

3.02727941

$3.02727941$ to find the turning point.

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Step 7.1

Find

f (3.02727941)

$f (3.02727941)$ to find the y-coordinate of

3.02727941

$3.02727941$ .

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Step 7.1.1

Replace the variable

x

$x$ with

3.02727941

$3.02727941$ in the expression.

f (3.02727941) = 4 {(3.02727941)}^{3} - {(3.02727941)}^{4} + 3.02727941 + 5

$f (3.02727941) = 4 {(3.02727941)}^{3} - {(3.02727941)}^{4} + 3.02727941 + 5$

Step 7.1.2

Simplify

4 {(3.02727941)}^{3} - {(3.02727941)}^{4} + 3.02727941 + 5

$4 {(3.02727941)}^{3} - {(3.02727941)}^{4} + 3.02727941 + 5$ .

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Step 7.1.2.1

Remove parentheses.

4 {(3.02727941)}^{3} - {(3.02727941)}^{4} + 3.02727941 + 5

$4 {(3.02727941)}^{3} - {(3.02727941)}^{4} + 3.02727941 + 5$

Step 7.1.2.2

Simplify each term.

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Step 7.1.2.2.1

Raise

3.02727941

$3.02727941$ to the power of

3

$3$ .

4 \cdot 27.7432619 - {(3.02727941)}^{4} + 3.02727941 + 5

$4 \cdot 27.7432619 - {(3.02727941)}^{4} + 3.02727941 + 5$

Step 7.1.2.2.2

Multiply

4

$4$ by

27.7432619

$27.7432619$ .

110.9730476 - {(3.02727941)}^{4} + 3.02727941 + 5

$110.9730476 - {(3.02727941)}^{4} + 3.02727941 + 5$

Step 7.1.2.2.3

Raise

3.02727941

$3.02727941$ to the power of

4

$4$ .

110.9730476 - 1 \cdot 83.98660555 + 3.02727941 + 5

$110.9730476 - 1 \cdot 83.98660555 + 3.02727941 + 5$

Step 7.1.2.2.4

Multiply

- 1

$- 1$ by

83.98660555

$83.98660555$ .

110.9730476 - 83.98660555 + 3.02727941 + 5

$110.9730476 - 83.98660555 + 3.02727941 + 5$

110.9730476 - 83.98660555 + 3.02727941 + 5

$110.9730476 - 83.98660555 + 3.02727941 + 5$

Step 7.1.2.3

Simplify by adding and subtracting.

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Step 7.1.2.3.1

Subtract

83.98660555

$83.98660555$ from

110.9730476

$110.9730476$ .

26.98644204 + 3.02727941 + 5

$26.98644204 + 3.02727941 + 5$

Step 7.1.2.3.2

Add

26.98644204

$26.98644204$ and

3.02727941

$3.02727941$ .

30.01372146 + 5

$30.01372146 + 5$

Step 7.1.2.3.3

Add

30.01372146

$30.01372146$ and

5

$5$ .

35.01372146

$35.01372146$

35.01372146

$35.01372146$

35.01372146

$35.01372146$

35.01372146

$35.01372146$

Step 7.2

Write the

x

$x$ and

y

$y$ coordinates in point form.

(3.02727941, 35.01372146)

$(3.02727941, 35.01372146)$

(3.02727941, 35.01372146)

$(3.02727941, 35.01372146)$

Step 8

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