Calculus Examples

$f (x) = \frac{1}{5} x^{5} + x^{4}$

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 $\frac{1}{5} x^{5} + x^{4}$ with respect to $x$ is $\frac{d}{d x} [\frac{1}{5} x^{5}] + \frac{d}{d x} [x^{4}]$ .

$\frac{d}{d x} [\frac{1}{5} x^{5}] + \frac{d}{d x} [x^{4}]$

Step 1.1.1.2

Evaluate $\frac{d}{d x} [\frac{1}{5} x^{5}]$ .

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Step 1.1.1.2.1

Since $\frac{1}{5}$ is constant with respect to $x$ , the derivative of $\frac{1}{5} x^{5}$ with respect to $x$ is $\frac{1}{5} \frac{d}{d x} [x^{5}]$ .

$\frac{1}{5} \frac{d}{d x} [x^{5}] + \frac{d}{d x} [x^{4}]$

Step 1.1.1.2.2

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

$\frac{1}{5} (5 x^{4}) + \frac{d}{d x} [x^{4}]$

Step 1.1.1.2.3

Combine $5$ and $\frac{1}{5}$ .

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

Step 1.1.1.2.4

Combine $\frac{5}{5}$ and $x^{4}$ .

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

Step 1.1.1.2.5

Cancel the common factor of $5$ .

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Step 1.1.1.2.5.1

Cancel the common factor.

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

Step 1.1.1.2.5.2

Divide $x^{4}$ by $1$ .

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

Step 1.1.1.3

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

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

Step 1.1.2

Find the second derivative.

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Step 1.1.2.1

Differentiate.

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Step 1.1.2.1.1

By the Sum Rule, the derivative of $x^{4} + 4 x^{3}$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [4 x^{3}]$ .

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

Step 1.1.2.1.2

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

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

Step 1.1.2.2

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

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Step 1.1.2.2.1

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

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

Step 1.1.2.2.2

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

$4 x^{3} + 4 (3 x^{2})$

Step 1.1.2.2.3

Multiply $3$ by $4$ .

$f'' (x) = 4 x^{3} + 12 x^{2}$

Step 1.1.3

The second derivative of $f (x)$ with respect to $x$ is $4 x^{3} + 12 x^{2}$ .

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

Step 1.2

Set the second derivative equal to $0$ then solve the equation $4 x^{3} + 12 x^{2} = 0$ .

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Step 1.2.1

Set the second derivative equal to $0$ .

$4 x^{3} + 12 x^{2} = 0$

Step 1.2.2

Factor $4 x^{2}$ out of $4 x^{3} + 12 x^{2}$ .

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Step 1.2.2.1

Factor $4 x^{2}$ out of $4 x^{3}$ .

$4 x^{2} (x) + 12 x^{2} = 0$

Step 1.2.2.2

Factor $4 x^{2}$ out of $12 x^{2}$ .

$4 x^{2} (x) + 4 x^{2} (3) = 0$

Step 1.2.2.3

Factor $4 x^{2}$ out of $4 x^{2} (x) + 4 x^{2} (3)$ .

$4 x^{2} (x + 3) = 0$

Step 1.2.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x^{2} = 0$

$x + 3 = 0$

Step 1.2.4

Set $x^{2}$ equal to $0$ and solve for $x$ .

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Step 1.2.4.1

Set $x^{2}$ equal to $0$ .

$x^{2} = 0$

Step 1.2.4.2

Solve $x^{2} = 0$ for $x$ .

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Step 1.2.4.2.1

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

$x = \pm \sqrt{0}$

Step 1.2.4.2.2

Simplify $\pm \sqrt{0}$ .

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Step 1.2.4.2.2.1

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

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

Step 1.2.4.2.2.2

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

$x = \pm 0$

Step 1.2.4.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 1.2.5

Set $x + 3$ equal to $0$ and solve for $x$ .

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Step 1.2.5.1

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Step 1.2.5.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 1.2.6

The final solution is all the values that make $4 x^{2} (x + 3) = 0$ true.

$x = 0, - 3$

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.

$(- \infty, - 3) \cup (- 3, 0) \cup (0, \infty)$

Step 4

Substitute any number from the interval $(- \infty, - 3)$ into the second derivative and evaluate to determine the concavity.

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Step 4.1

Replace the variable $x$ with $- 6$ in the expression.

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

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

Raise $- 6$ to the power of $3$ .

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

Step 4.2.1.2

Multiply $4$ by $- 216$ .

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

Step 4.2.1.3

Raise $- 6$ to the power of $2$ .

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

Step 4.2.1.4

Multiply $12$ by $36$ .

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

Step 4.2.2

Add $- 864$ and $432$ .

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

Step 4.2.3

The final answer is $- 432$ .

$- 432$

Step 4.3

The graph is concave down on the interval $(- \infty, - 3)$ because $f'' (- 6)$ is negative.

Concave down on $(- \infty, - 3)$ since $f'' (x)$ is negative

Step 5

Substitute any number from the interval $(- 3, 0)$ 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.

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

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

$f'' (- 2) = 4 \cdot - 8 + 12 {(- 2)}^{2}$

Step 5.2.1.2

Multiply $4$ by $- 8$ .

$f'' (- 2) = - 32 + 12 {(- 2)}^{2}$

Step 5.2.1.3

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

$f'' (- 2) = - 32 + 12 \cdot 4$

Step 5.2.1.4

Multiply $12$ by $4$ .

$f'' (- 2) = - 32 + 48$

Step 5.2.2

Add $- 32$ and $48$ .

$f'' (- 2) = 16$

Step 5.2.3

The final answer is $16$ .

$16$

Step 5.3

The graph is concave up on the interval $(- 3, 0)$ because $f'' (- 2)$ is positive.

Concave up on $(- 3, 0)$ since $f'' (x)$ is positive

Step 6

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

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Step 6.1

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

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

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

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

$f'' (2) = 4 \cdot 8 + 12 {(2)}^{2}$

Step 6.2.1.2

Multiply $4$ by $8$ .

$f'' (2) = 32 + 12 {(2)}^{2}$

Step 6.2.1.3

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

$f'' (2) = 32 + 12 \cdot 4$

Step 6.2.1.4

Multiply $12$ by $4$ .

$f'' (2) = 32 + 48$

Step 6.2.2

Add $32$ and $48$ .

$f'' (2) = 80$

Step 6.2.3

The final answer is $80$ .

$80$

Step 6.3

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 7

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

Concave down on $(- \infty, - 3)$ since $f'' (x)$ is negative

Concave up on $(- 3, 0)$ since $f'' (x)$ is positive

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

Step 8