Calculus Examples

$\frac{1}{3} x^{3} - 3 x^{2} - 16 x$

Step 1

Write $\frac{1}{3} x^{3} - 3 x^{2} - 16 x$ as a function.

$f (x) = \frac{1}{3} x^{3} - 3 x^{2} - 16 x$

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 $\frac{1}{3} x^{3} - 3 x^{2} - 16 x$ with respect to $x$ is $\frac{d}{d x} [\frac{1}{3} x^{3}] + \frac{d}{d x} [- 3 x^{2}] + \frac{d}{d x} [- 16 x]$ .

$\frac{d}{d x} [\frac{1}{3} x^{3}] + \frac{d}{d x} [- 3 x^{2}] + \frac{d}{d x} [- 16 x]$

Step 2.1.1.2

Evaluate $\frac{d}{d x} [\frac{1}{3} x^{3}]$ .

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Step 2.1.1.2.1

Since $\frac{1}{3}$ is constant with respect to $x$ , the derivative of $\frac{1}{3} x^{3}$ with respect to $x$ is $\frac{1}{3} \frac{d}{d x} [x^{3}]$ .

$\frac{1}{3} \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 3 x^{2}] + \frac{d}{d x} [- 16 x]$

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 $n = 3$ .

$\frac{1}{3} (3 x^{2}) + \frac{d}{d x} [- 3 x^{2}] + \frac{d}{d x} [- 16 x]$

Step 2.1.1.2.3

Combine $3$ and $\frac{1}{3}$ .

$\frac{3}{3} x^{2} + \frac{d}{d x} [- 3 x^{2}] + \frac{d}{d x} [- 16 x]$

Step 2.1.1.2.4

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

$\frac{3 x^{2}}{3} + \frac{d}{d x} [- 3 x^{2}] + \frac{d}{d x} [- 16 x]$

Step 2.1.1.2.5

Cancel the common factor of $3$ .

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Step 2.1.1.2.5.1

Cancel the common factor.

$\frac{3 x^{2}}{3} + \frac{d}{d x} [- 3 x^{2}] + \frac{d}{d x} [- 16 x]$

Step 2.1.1.2.5.2

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

$x^{2} + \frac{d}{d x} [- 3 x^{2}] + \frac{d}{d x} [- 16 x]$

Step 2.1.1.3

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

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Step 2.1.1.3.1

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

$x^{2} - 3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 16 x]$

Step 2.1.1.3.2

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

$x^{2} - 3 (2 x) + \frac{d}{d x} [- 16 x]$

Step 2.1.1.3.3

Multiply $2$ by $- 3$ .

$x^{2} - 6 x + \frac{d}{d x} [- 16 x]$

Step 2.1.1.4

Evaluate $\frac{d}{d x} [- 16 x]$ .

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Step 2.1.1.4.1

Since $- 16$ is constant with respect to $x$ , the derivative of $- 16 x$ with respect to $x$ is $- 16 \frac{d}{d x} [x]$ .

$x^{2} - 6 x - 16 \frac{d}{d x} [x]$

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 $n = 1$ .

$x^{2} - 6 x - 16 \cdot 1$

Step 2.1.1.4.3

Multiply $- 16$ by $1$ .

$f' (x) = x^{2} - 6 x - 16$

Step 2.1.2

Find the second derivative.

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Step 2.1.2.1

Differentiate.

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Step 2.1.2.1.1

By the Sum Rule, the derivative of $x^{2} - 6 x - 16$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [- 16]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [- 16]$

Step 2.1.2.1.2

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

$2 x + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [- 16]$

Step 2.1.2.2

Evaluate $\frac{d}{d x} [- 6 x]$ .

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Step 2.1.2.2.1

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

$2 x - 6 \frac{d}{d x} [x] + \frac{d}{d x} [- 16]$

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 $n = 1$ .

$2 x - 6 \cdot 1 + \frac{d}{d x} [- 16]$

Step 2.1.2.2.3

Multiply $- 6$ by $1$ .

$2 x - 6 + \frac{d}{d x} [- 16]$

Step 2.1.2.3

Differentiate using the Constant Rule.

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Step 2.1.2.3.1

Since $- 16$ is constant with respect to $x$ , the derivative of $- 16$ with respect to $x$ is $0$ .

$2 x - 6 + 0$

Step 2.1.2.3.2

Add $2 x - 6$ and $0$ .

$f'' (x) = 2 x - 6$

Step 2.1.3

The second derivative of $f (x)$ with respect to $x$ is $2 x - 6$ .

$2 x - 6$

Step 2.2

Set the second derivative equal to $0$ then solve the equation $2 x - 6 = 0$ .

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Step 2.2.1

Set the second derivative equal to $0$ .

$2 x - 6 = 0$

Step 2.2.2

Add $6$ to both sides of the equation.

$2 x = 6$

Step 2.2.3

Divide each term in $2 x = 6$ by $2$ and simplify.

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Step 2.2.3.1

Divide each term in $2 x = 6$ by $2$ .

$\frac{2 x}{2} = \frac{6}{2}$

Step 2.2.3.2

Simplify the left side.

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Step 2.2.3.2.1

Cancel the common factor of $2$ .

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Step 2.2.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{6}{2}$

Step 2.2.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{6}{2}$

Step 2.2.3.3

Simplify the right side.

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Step 2.2.3.3.1

Divide $6$ by $2$ .

$x = 3$

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.

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

Step 5

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

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Step 5.1

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

$f'' (0) = 2 (0) - 6$

Step 5.2

Simplify the result.

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Step 5.2.1

Multiply $2$ by $0$ .

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

Step 5.2.2

Subtract $6$ from $0$ .

$f'' (0) = - 6$

Step 5.2.3

The final answer is $- 6$ .

$- 6$

Step 5.3

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

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

Step 6

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

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Step 6.1

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

$f'' (6) = 2 (6) - 6$

Step 6.2

Simplify the result.

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Step 6.2.1

Multiply $2$ by $6$ .

$f'' (6) = 12 - 6$

Step 6.2.2

Subtract $6$ from $12$ .

$f'' (6) = 6$

Step 6.2.3

The final answer is $6$ .

$6$

Step 6.3

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

Step 8