Calculus Examples

$f (x) = x^{5} - 8$

Step 1

Find the $x$ values where the second derivative is equal to $0$ .

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Step 1.1

Find the second derivative.

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Step 1.1.1

Find the first derivative.

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Step 1.1.1.1

By the Sum Rule, the derivative of $x^{5} - 8$ with respect to $x$ is $\frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 8]$ .

$\frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 8]$

Step 1.1.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$5 x^{4} + \frac{d}{d x} [- 8]$

Step 1.1.1.3

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8$ with respect to $x$ is $0$ .

$5 x^{4} + 0$

Step 1.1.1.4

Add $5 x^{4}$ and $0$ .

$f' (x) = 5 x^{4}$

Step 1.1.2

Find the second derivative.

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Step 1.1.2.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{4}$ with respect to $x$ is $5 \frac{d}{d x} [x^{4}]$ .

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

Step 1.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$5 (4 x^{3})$

Step 1.1.2.3

Multiply $4$ by $5$ .

$f'' (x) = 20 x^{3}$

Step 1.1.3

The second derivative of $f (x)$ with respect to $x$ is $20 x^{3}$ .

$20 x^{3}$

Step 1.2

Set the second derivative equal to $0$ then solve the equation $20 x^{3} = 0$ .

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Step 1.2.1

Set the second derivative equal to $0$ .

$20 x^{3} = 0$

Step 1.2.2

Divide each term in $20 x^{3} = 0$ by $20$ and simplify.

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Step 1.2.2.1

Divide each term in $20 x^{3} = 0$ by $20$ .

$\frac{20 x^{3}}{20} = \frac{0}{20}$

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Cancel the common factor of $20$ .

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Step 1.2.2.2.1.1

Cancel the common factor.

$\frac{20 x^{3}}{20} = \frac{0}{20}$

Step 1.2.2.2.1.2

Divide $x^{3}$ by $1$ .

$x^{3} = \frac{0}{20}$

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Divide $0$ by $20$ .

$x^{3} = 0$

Step 1.2.3

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

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

Step 1.2.4

Simplify $\sqrt[3]{0}$ .

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Step 1.2.4.1

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

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

Step 1.2.4.2

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

$x = 0$

Step 2

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 3

Create intervals around the $x$ -values where the second derivative is zero or undefined.

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

Step 4

Substitute any number from the interval $(- \infty, 0)$ into the second derivative and evaluate to determine the concavity.

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Step 4.1

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

$f'' (- 2) = 20 {(- 2)}^{3}$

Step 4.2

Simplify the result.

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Step 4.2.1

Raise $- 2$ to the power of $3$ .

$f'' (- 2) = 20 \cdot - 8$

Step 4.2.2

Multiply $20$ by $- 8$ .

$f'' (- 2) = - 160$

Step 4.2.3

The final answer is $- 160$ .

$- 160$

Step 4.3

The graph is concave down on the interval $(- \infty, 0)$ because $f'' (- 2)$ is negative.

Concave down on $(- \infty, 0)$ since $f'' (x)$ is negative

Step 5

Substitute any number from the interval $(0, \infty)$ into the second derivative and evaluate to determine the concavity.

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Step 5.1

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

$f'' (2) = 20 {(2)}^{3}$

Step 5.2

Simplify the result.

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Step 5.2.1

Raise $2$ to the power of $3$ .

$f'' (2) = 20 \cdot 8$

Step 5.2.2

Multiply $20$ by $8$ .

$f'' (2) = 160$

Step 5.2.3

The final answer is $160$ .

$160$

Step 5.3

The graph is concave up on the interval $(0, \infty)$ because $f'' (2)$ is positive.

Concave up on $(0, \infty)$ since $f'' (x)$ is positive

Step 6

The graph is concave down when the second derivative is negative and concave up when the second derivative is positive.

Concave down on $(- \infty, 0)$ since $f'' (x)$ is negative

Concave up on $(0, \infty)$ since $f'' (x)$ is positive

Step 7

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