Calculus Examples

$f (x) = - \frac{1}{5} x^{5} - 3 x^{4} - 12 x^{3}$

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 $- \frac{1}{5} x^{5} - 3 x^{4} - 12 x^{3}$ with respect to $x$ is $\frac{d}{d x} [- \frac{1}{5} x^{5}] + \frac{d}{d x} [- 3 x^{4}] + \frac{d}{d x} [- 12 x^{3}]$ .

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

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

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

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 $n = 5$ .

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

Step 1.1.1.2.3

Multiply $5$ by $- 1$ .

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

Step 1.1.1.2.4

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

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

Step 1.1.1.2.5

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

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

Step 1.1.1.2.6

Cancel the common factor of $- 5$ and $5$ .

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Step 1.1.1.2.6.1

Factor $5$ out of $- 5 x^{4}$ .

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

Step 1.1.1.2.6.2

Cancel the common factors.

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Step 1.1.1.2.6.2.1

Factor $5$ out of $5$ .

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

Step 1.1.1.2.6.2.2

Cancel the common factor.

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

Step 1.1.1.2.6.2.3

Rewrite the expression.

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

Step 1.1.1.2.6.2.4

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

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

Step 1.1.1.3

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

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Step 1.1.1.3.1

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

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

Step 1.1.1.3.2

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

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

Step 1.1.1.3.3

Multiply $4$ by $- 3$ .

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

Step 1.1.1.4

Evaluate $\frac{d}{d x} [- 12 x^{3}]$ .

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Step 1.1.1.4.1

Since $- 12$ is constant with respect to $x$ , the derivative of $- 12 x^{3}$ with respect to $x$ is $- 12 \frac{d}{d x} [x^{3}]$ .

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

Step 1.1.1.4.2

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

$- x^{4} - 12 x^{3} - 12 (3 x^{2})$

Step 1.1.1.4.3

Multiply $3$ by $- 12$ .

$f' (x) = - x^{4} - 12 x^{3} - 36 x^{2}$

Step 1.1.2

Find the second derivative.

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Step 1.1.2.1

By the Sum Rule, the derivative of $- x^{4} - 12 x^{3} - 36 x^{2}$ with respect to $x$ is $\frac{d}{d x} [- x^{4}] + \frac{d}{d x} [- 12 x^{3}] + \frac{d}{d x} [- 36 x^{2}]$ .

$\frac{d}{d x} [- x^{4}] + \frac{d}{d x} [- 12 x^{3}] + \frac{d}{d x} [- 36 x^{2}]$

Step 1.1.2.2

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

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Step 1.1.2.2.1

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

$- \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 12 x^{3}] + \frac{d}{d x} [- 36 x^{2}]$

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 $n = 4$ .

$- (4 x^{3}) + \frac{d}{d x} [- 12 x^{3}] + \frac{d}{d x} [- 36 x^{2}]$

Step 1.1.2.2.3

Multiply $4$ by $- 1$ .

$- 4 x^{3} + \frac{d}{d x} [- 12 x^{3}] + \frac{d}{d x} [- 36 x^{2}]$

Step 1.1.2.3

Evaluate $\frac{d}{d x} [- 12 x^{3}]$ .

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Step 1.1.2.3.1

Since $- 12$ is constant with respect to $x$ , the derivative of $- 12 x^{3}$ with respect to $x$ is $- 12 \frac{d}{d x} [x^{3}]$ .

$- 4 x^{3} - 12 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 36 x^{2}]$

Step 1.1.2.3.2

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

$- 4 x^{3} - 12 (3 x^{2}) + \frac{d}{d x} [- 36 x^{2}]$

Step 1.1.2.3.3

Multiply $3$ by $- 12$ .

$- 4 x^{3} - 36 x^{2} + \frac{d}{d x} [- 36 x^{2}]$

Step 1.1.2.4

Evaluate $\frac{d}{d x} [- 36 x^{2}]$ .

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Step 1.1.2.4.1

Since $- 36$ is constant with respect to $x$ , the derivative of $- 36 x^{2}$ with respect to $x$ is $- 36 \frac{d}{d x} [x^{2}]$ .

$- 4 x^{3} - 36 x^{2} - 36 \frac{d}{d x} [x^{2}]$

Step 1.1.2.4.2

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

$- 4 x^{3} - 36 x^{2} - 36 (2 x)$

Step 1.1.2.4.3

Multiply $2$ by $- 36$ .

$f'' (x) = - 4 x^{3} - 36 x^{2} - 72 x$

Step 1.1.3

The second derivative of $f (x)$ with respect to $x$ is $- 4 x^{3} - 36 x^{2} - 72 x$ .

$- 4 x^{3} - 36 x^{2} - 72 x$

Step 1.2

Set the second derivative equal to $0$ then solve the equation $- 4 x^{3} - 36 x^{2} - 72 x = 0$ .

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Step 1.2.1

Set the second derivative equal to $0$ .

$- 4 x^{3} - 36 x^{2} - 72 x = 0$

Step 1.2.2

Factor the left side of the equation.

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Step 1.2.2.1

Factor $- 4 x$ out of $- 4 x^{3} - 36 x^{2} - 72 x$ .

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Step 1.2.2.1.1

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

$- 4 x \cdot x^{2} - 36 x^{2} - 72 x = 0$

Step 1.2.2.1.2

Factor $- 4 x$ out of $- 36 x^{2}$ .

$- 4 x \cdot x^{2} - 4 x (9 x) - 72 x = 0$

Step 1.2.2.1.3

Factor $- 4 x$ out of $- 72 x$ .

$- 4 x \cdot x^{2} - 4 x (9 x) - 4 x \cdot 18 = 0$

Step 1.2.2.1.4

Factor $- 4 x$ out of $- 4 x (x^{2}) - 4 x (9 x)$ .

$- 4 x (x^{2} + 9 x) - 4 x \cdot 18 = 0$

Step 1.2.2.1.5

Factor $- 4 x$ out of $- 4 x (x^{2} + 9 x) - 4 x (18)$ .

$- 4 x (x^{2} + 9 x + 18) = 0$

Step 1.2.2.2

Factor.

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Step 1.2.2.2.1

Factor $x^{2} + 9 x + 18$ using the AC method.

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Step 1.2.2.2.1.1

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 $9$ .

$3, 6$

Step 1.2.2.2.1.2

Write the factored form using these integers.

$- 4 x ((x + 3) (x + 6)) = 0$

Step 1.2.2.2.2

Remove unnecessary parentheses.

$- 4 x (x + 3) (x + 6) = 0$

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 $0$ .

$x = 0$

$x + 3 = 0$

$x + 6 = 0$

Step 1.2.4

Set $x$ equal to $0$ .

$x = 0$

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

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

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Step 1.2.6.1

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

$x + 6 = 0$

Step 1.2.6.2

Subtract $6$ from both sides of the equation.

$x = - 6$

Step 1.2.7

The final solution is all the values that make $- 4 x (x + 3) (x + 6) = 0$ true.

$x = 0, - 3, - 6$

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.

$(- \infty, - 6) \cup (- 6, - 3) \cup (- 3, 0) \cup (0, \infty)$

Step 4

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

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Step 4.1

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

$f'' (- 8) = - 4 {(- 8)}^{3} - 36 {(- 8)}^{2} - 72 \cdot - 8$

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 $- 8$ to the power of $3$ .

$f'' (- 8) = - 4 \cdot - 512 - 36 {(- 8)}^{2} - 72 \cdot - 8$

Step 4.2.1.2

Multiply $- 4$ by $- 512$ .

$f'' (- 8) = 2048 - 36 {(- 8)}^{2} - 72 \cdot - 8$

Step 4.2.1.3

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

$f'' (- 8) = 2048 - 36 \cdot 64 - 72 \cdot - 8$

Step 4.2.1.4

Multiply $- 36$ by $64$ .

$f'' (- 8) = 2048 - 2304 - 72 \cdot - 8$

Step 4.2.1.5

Multiply $- 72$ by $- 8$ .

$f'' (- 8) = 2048 - 2304 + 576$

Step 4.2.2

Simplify by adding and subtracting.

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Step 4.2.2.1

Subtract $2304$ from $2048$ .

$f'' (- 8) = - 256 + 576$

Step 4.2.2.2

Add $- 256$ and $576$ .

$f'' (- 8) = 320$

Step 4.2.3

The final answer is $320$ .

$320$

Step 4.3

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 5

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

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Step 5.1

Replace the variable $x$ with $- 4$ in the expression.

$f'' (- 4) = - 4 {(- 4)}^{3} - 36 {(- 4)}^{2} - 72 \cdot - 4$

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

Multiply $- 4$ by ${(- 4)}^{3}$ by adding the exponents.

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Step 5.2.1.1.1

Multiply $- 4$ by ${(- 4)}^{3}$ .

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Step 5.2.1.1.1.1

Raise $- 4$ to the power of $1$ .

$f'' (- 4) = (- 4) {(- 4)}^{3} - 36 {(- 4)}^{2} - 72 \cdot - 4$

Step 5.2.1.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (- 4) = {(- 4)}^{1 + 3} - 36 {(- 4)}^{2} - 72 \cdot - 4$

Step 5.2.1.1.2

Add $1$ and $3$ .

$f'' (- 4) = {(- 4)}^{4} - 36 {(- 4)}^{2} - 72 \cdot - 4$

Step 5.2.1.2

Raise $- 4$ to the power of $4$ .

$f'' (- 4) = 256 - 36 {(- 4)}^{2} - 72 \cdot - 4$

Step 5.2.1.3

Raise $- 4$ to the power of $2$ .

$f'' (- 4) = 256 - 36 \cdot 16 - 72 \cdot - 4$

Step 5.2.1.4

Multiply $- 36$ by $16$ .

$f'' (- 4) = 256 - 576 - 72 \cdot - 4$

Step 5.2.1.5

Multiply $- 72$ by $- 4$ .

$f'' (- 4) = 256 - 576 + 288$

Step 5.2.2

Simplify by adding and subtracting.

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Step 5.2.2.1

Subtract $576$ from $256$ .

$f'' (- 4) = - 320 + 288$

Step 5.2.2.2

Add $- 320$ and $288$ .

$f'' (- 4) = - 32$

Step 5.2.3

The final answer is $- 32$ .

$- 32$

Step 5.3

The graph is concave down on the interval $(- 6, - 3)$ because $f'' (- 4)$ is negative.

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

Step 6

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

$f'' (- 2) = - 4 {(- 2)}^{3} - 36 {(- 2)}^{2} - 72 \cdot - 2$

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 $3$ .

$f'' (- 2) = - 4 \cdot - 8 - 36 {(- 2)}^{2} - 72 \cdot - 2$

Step 6.2.1.2

Multiply $- 4$ by $- 8$ .

$f'' (- 2) = 32 - 36 {(- 2)}^{2} - 72 \cdot - 2$

Step 6.2.1.3

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

$f'' (- 2) = 32 - 36 \cdot 4 - 72 \cdot - 2$

Step 6.2.1.4

Multiply $- 36$ by $4$ .

$f'' (- 2) = 32 - 144 - 72 \cdot - 2$

Step 6.2.1.5

Multiply $- 72$ by $- 2$ .

$f'' (- 2) = 32 - 144 + 144$

Step 6.2.2

Simplify by adding and subtracting.

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Step 6.2.2.1

Subtract $144$ from $32$ .

$f'' (- 2) = - 112 + 144$

Step 6.2.2.2

Add $- 112$ and $144$ .

$f'' (- 2) = 32$

Step 6.2.3

The final answer is $32$ .

$32$

Step 6.3

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 7

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

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Step 7.1

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

$f'' (2) = - 4 {(2)}^{3} - 36 {(2)}^{2} - 72 \cdot 2$

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 $2$ to the power of $3$ .

$f'' (2) = - 4 \cdot 8 - 36 {(2)}^{2} - 72 \cdot 2$

Step 7.2.1.2

Multiply $- 4$ by $8$ .

$f'' (2) = - 32 - 36 {(2)}^{2} - 72 \cdot 2$

Step 7.2.1.3

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

$f'' (2) = - 32 - 36 \cdot 4 - 72 \cdot 2$

Step 7.2.1.4

Multiply $- 36$ by $4$ .

$f'' (2) = - 32 - 144 - 72 \cdot 2$

Step 7.2.1.5

Multiply $- 72$ by $2$ .

$f'' (2) = - 32 - 144 - 144$

Step 7.2.2

Simplify by subtracting numbers.

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Step 7.2.2.1

Subtract $144$ from $- 32$ .

$f'' (2) = - 176 - 144$

Step 7.2.2.2

Subtract $144$ from $- 176$ .

$f'' (2) = - 320$

Step 7.2.3

The final answer is $- 320$ .

$- 320$

Step 7.3

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

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

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

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

Step 9