Calculus Examples

$\frac{1}{6} x^{4} + x^{3} - 18 x^{2}$

Step 1

Write $\frac{1}{6} x^{4} + x^{3} - 18 x^{2}$ as a function.

$f (x) = \frac{1}{6} x^{4} + x^{3} - 18 x^{2}$

Step 2

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

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Step 2.1

Find the second derivative.

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Step 2.1.1

Find the first derivative.

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Step 2.1.1.1

By the Sum Rule, the derivative of $\frac{1}{6} x^{4} + x^{3} - 18 x^{2}$ with respect to $x$ is $\frac{d}{d x} [\frac{1}{6} x^{4}] + \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 18 x^{2}]$ .

$\frac{d}{d x} [\frac{1}{6} x^{4}] + \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 18 x^{2}]$

Step 2.1.1.2

Evaluate $\frac{d}{d x} [\frac{1}{6} x^{4}]$ .

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Step 2.1.1.2.1

Since $\frac{1}{6}$ is constant with respect to $x$ , the derivative of $\frac{1}{6} x^{4}$ with respect to $x$ is $\frac{1}{6} \frac{d}{d x} [x^{4}]$ .

$\frac{1}{6} \frac{d}{d x} [x^{4}] + \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 18 x^{2}]$

Step 2.1.1.2.2

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

$\frac{1}{6} (4 x^{3}) + \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 18 x^{2}]$

Step 2.1.1.2.3

Combine $4$ and $\frac{1}{6}$ .

$\frac{4}{6} x^{3} + \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 18 x^{2}]$

Step 2.1.1.2.4

Combine $\frac{4}{6}$ and $x^{3}$ .

$\frac{4 x^{3}}{6} + \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 18 x^{2}]$

Step 2.1.1.2.5

Cancel the common factor of $4$ and $6$ .

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Step 2.1.1.2.5.1

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

$\frac{2 (2 x^{3})}{6} + \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 18 x^{2}]$

Step 2.1.1.2.5.2

Cancel the common factors.

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Step 2.1.1.2.5.2.1

Factor $2$ out of $6$ .

$\frac{2 (2 x^{3})}{2 (3)} + \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 18 x^{2}]$

Step 2.1.1.2.5.2.2

Cancel the common factor.

$\frac{2 (2 x^{3})}{2 \cdot 3} + \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 18 x^{2}]$

Step 2.1.1.2.5.2.3

Rewrite the expression.

$\frac{2 x^{3}}{3} + \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 18 x^{2}]$

Step 2.1.1.3

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

$\frac{2 x^{3}}{3} + 3 x^{2} + \frac{d}{d x} [- 18 x^{2}]$

Step 2.1.1.4

Evaluate $\frac{d}{d x} [- 18 x^{2}]$ .

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Step 2.1.1.4.1

Since $- 18$ is constant with respect to $x$ , the derivative of $- 18 x^{2}$ with respect to $x$ is $- 18 \frac{d}{d x} [x^{2}]$ .

$\frac{2 x^{3}}{3} + 3 x^{2} - 18 \frac{d}{d x} [x^{2}]$

Step 2.1.1.4.2

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

$\frac{2 x^{3}}{3} + 3 x^{2} - 18 (2 x)$

Step 2.1.1.4.3

Multiply $2$ by $- 18$ .

$f' (x) = \frac{2 x^{3}}{3} + 3 x^{2} - 36 x$

Step 2.1.2

Find the second derivative.

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Step 2.1.2.1

By the Sum Rule, the derivative of $\frac{2 x^{3}}{3} + 3 x^{2} - 36 x$ with respect to $x$ is $\frac{d}{d x} [\frac{2 x^{3}}{3}] + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 36 x]$ .

$\frac{d}{d x} [\frac{2 x^{3}}{3}] + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 36 x]$

Step 2.1.2.2

Evaluate $\frac{d}{d x} [\frac{2 x^{3}}{3}]$ .

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Step 2.1.2.2.1

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

$\frac{2}{3} \frac{d}{d x} [x^{3}] + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 36 x]$

Step 2.1.2.2.2

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

$\frac{2}{3} (3 x^{2}) + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 36 x]$

Step 2.1.2.2.3

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

$\frac{3 \cdot 2}{3} x^{2} + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 36 x]$

Step 2.1.2.2.4

Multiply $3$ by $2$ .

$\frac{6}{3} x^{2} + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 36 x]$

Step 2.1.2.2.5

Combine $\frac{6}{3}$ and $x^{2}$ .

$\frac{6 x^{2}}{3} + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 36 x]$

Step 2.1.2.2.6

Cancel the common factor of $6$ and $3$ .

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Step 2.1.2.2.6.1

Factor $3$ out of $6 x^{2}$ .

$\frac{3 (2 x^{2})}{3} + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 36 x]$

Step 2.1.2.2.6.2

Cancel the common factors.

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Step 2.1.2.2.6.2.1

Factor $3$ out of $3$ .

$\frac{3 (2 x^{2})}{3 (1)} + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 36 x]$

Step 2.1.2.2.6.2.2

Cancel the common factor.

$\frac{3 (2 x^{2})}{3 \cdot 1} + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 36 x]$

Step 2.1.2.2.6.2.3

Rewrite the expression.

$\frac{2 x^{2}}{1} + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 36 x]$

Step 2.1.2.2.6.2.4

Divide $2 x^{2}$ by $1$ .

$2 x^{2} + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 36 x]$

Step 2.1.2.3

Evaluate $\frac{d}{d x} [3 x^{2}]$ .

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Step 2.1.2.3.1

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

$2 x^{2} + 3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 36 x]$

Step 2.1.2.3.2

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

$2 x^{2} + 3 (2 x) + \frac{d}{d x} [- 36 x]$

Step 2.1.2.3.3

Multiply $2$ by $3$ .

$2 x^{2} + 6 x + \frac{d}{d x} [- 36 x]$

Step 2.1.2.4

Evaluate $\frac{d}{d x} [- 36 x]$ .

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Step 2.1.2.4.1

Since $- 36$ is constant with respect to $x$ , the derivative of $- 36 x$ with respect to $x$ is $- 36 \frac{d}{d x} [x]$ .

$2 x^{2} + 6 x - 36 \frac{d}{d x} [x]$

Step 2.1.2.4.2

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

$2 x^{2} + 6 x - 36 \cdot 1$

Step 2.1.2.4.3

Multiply $- 36$ by $1$ .

$f'' (x) = 2 x^{2} + 6 x - 36$

Step 2.1.3

The second derivative of $f (x)$ with respect to $x$ is $2 x^{2} + 6 x - 36$ .

$2 x^{2} + 6 x - 36$

Step 2.2

Set the second derivative equal to $0$ then solve the equation $2 x^{2} + 6 x - 36 = 0$ .

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Step 2.2.1

Set the second derivative equal to $0$ .

$2 x^{2} + 6 x - 36 = 0$

Step 2.2.2

Factor the left side of the equation.

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Step 2.2.2.1

Factor $2$ out of $2 x^{2} + 6 x - 36$ .

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Step 2.2.2.1.1

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

$2 (x^{2}) + 6 x - 36 = 0$

Step 2.2.2.1.2

Factor $2$ out of $6 x$ .

$2 (x^{2}) + 2 (3 x) - 36 = 0$

Step 2.2.2.1.3

Factor $2$ out of $- 36$ .

$2 x^{2} + 2 (3 x) + 2 \cdot - 18 = 0$

Step 2.2.2.1.4

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

$2 (x^{2} + 3 x) + 2 \cdot - 18 = 0$

Step 2.2.2.1.5

Factor $2$ out of $2 (x^{2} + 3 x) + 2 \cdot - 18$ .

$2 (x^{2} + 3 x - 18) = 0$

Step 2.2.2.2

Factor.

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Step 2.2.2.2.1

Factor $x^{2} + 3 x - 18$ using the AC method.

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Step 2.2.2.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 18$ and whose sum is $3$ .

$- 3, 6$

Step 2.2.2.2.1.2

Write the factored form using these integers.

$2 ((x - 3) (x + 6)) = 0$

Step 2.2.2.2.2

Remove unnecessary parentheses.

$2 (x - 3) (x + 6) = 0$

Step 2.2.3

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

$x - 3 = 0$

$x + 6 = 0$

Step 2.2.4

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

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Step 2.2.4.1

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

$x - 3 = 0$

Step 2.2.4.2

Add $3$ to both sides of the equation.

$x = 3$

Step 2.2.5

Set $x + 6$ equal to $0$ and solve for $x$ .

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Step 2.2.5.1

Set $x + 6$ equal to $0$ .

$x + 6 = 0$

Step 2.2.5.2

Subtract $6$ from both sides of the equation.

$x = - 6$

Step 2.2.6

The final solution is all the values that make $2 (x - 3) (x + 6) = 0$ true.

$x = 3, - 6$

Step 3

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 4

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

$(- \infty, - 6) \cup (- 6, 3) \cup (3, \infty)$

Step 5

Substitute any number from the interval $(- \infty, - 6)$ into the second derivative and evaluate to determine the concavity.

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Step 5.1

Replace the variable $x$ with $- 8$ in the expression.

$f'' (- 8) = 2 {(- 8)}^{2} + 6 (- 8) - 36$

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 $- 8$ to the power of $2$ .

$f'' (- 8) = 2 \cdot 64 + 6 (- 8) - 36$

Step 5.2.1.2

Multiply $2$ by $64$ .

$f'' (- 8) = 128 + 6 (- 8) - 36$

Step 5.2.1.3

Multiply $6$ by $- 8$ .

$f'' (- 8) = 128 - 48 - 36$

Step 5.2.2

Simplify by subtracting numbers.

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Step 5.2.2.1

Subtract $48$ from $128$ .

$f'' (- 8) = 80 - 36$

Step 5.2.2.2

Subtract $36$ from $80$ .

$f'' (- 8) = 44$

Step 5.2.3

The final answer is $44$ .

$44$

Step 5.3

The graph is concave up on the interval $(- \infty, - 6)$ because $f'' (- 8)$ is positive.

Concave up on $(- \infty, - 6)$ since $f'' (x)$ is positive

Step 6

Substitute any number from the interval $(- 6, 3)$ into the second derivative and evaluate to determine the concavity.

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Step 6.1

Replace the variable $x$ with $0$ in the expression.

$f'' (0) = 2 {(0)}^{2} + 6 (0) - 36$

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

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

$f'' (0) = 2 \cdot 0 + 6 (0) - 36$

Step 6.2.1.2

Multiply $2$ by $0$ .

$f'' (0) = 0 + 6 (0) - 36$

Step 6.2.1.3

Multiply $6$ by $0$ .

$f'' (0) = 0 + 0 - 36$

Step 6.2.2

Simplify by adding and subtracting.

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Step 6.2.2.1

Add $0$ and $0$ .

$f'' (0) = 0 - 36$

Step 6.2.2.2

Subtract $36$ from $0$ .

$f'' (0) = - 36$

Step 6.2.3

The final answer is $- 36$ .

$- 36$

Step 6.3

The graph is concave down on the interval $(- 6, 3)$ because $f'' (0)$ is negative.

Concave down on $(- 6, 3)$ since $f'' (x)$ is negative

Step 7

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

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Step 7.1

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

$f'' (6) = 2 {(6)}^{2} + 6 (6) - 36$

Step 7.2

Simplify the result.

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Step 7.2.1

Simplify each term.

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Step 7.2.1.1

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

$f'' (6) = 2 \cdot 36 + 6 (6) - 36$

Step 7.2.1.2

Multiply $2$ by $36$ .

$f'' (6) = 72 + 6 (6) - 36$

Step 7.2.1.3

Multiply $6$ by $6$ .

$f'' (6) = 72 + 36 - 36$

Step 7.2.2

Simplify by adding and subtracting.

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Step 7.2.2.1

Add $72$ and $36$ .

$f'' (6) = 108 - 36$

Step 7.2.2.2

Subtract $36$ from $108$ .

$f'' (6) = 72$

Step 7.2.3

The final answer is $72$ .

$72$

Step 7.3

The graph is concave up on the interval $(3, \infty)$ because $f'' (6)$ is positive.

Concave up on $(3, \infty)$ since $f'' (x)$ is positive

Step 8

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

Concave up on $(- \infty, - 6)$ since $f'' (x)$ is positive

Concave down on $(- 6, 3)$ since $f'' (x)$ is negative

Concave up on $(3, \infty)$ since $f'' (x)$ is positive

Step 9