Calculus Examples

$f (x) = 2 x^{3} + 3 x^{2} - 336 x$

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 $2 x^{3} + 3 x^{2} - 336 x$ with respect to $x$ is $\frac{d}{d x} [2 x^{3}] + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 336 x]$ .

$f' (x) = \frac{d}{d x} (2 x^{3}) + \frac{d}{d x} (3 x^{2}) + \frac{d}{d x} (- 336 x)$

Step 1.1.1.2

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

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Step 1.1.1.2.1

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

$f' (x) = 2 \frac{d}{d x} (x^{3}) + \frac{d}{d x} (3 x^{2}) + \frac{d}{d x} (- 336 x)$

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 $n = 3$ .

$f' (x) = 2 (3 x^{2}) + \frac{d}{d x} (3 x^{2}) + \frac{d}{d x} (- 336 x)$

Step 1.1.1.2.3

Multiply $3$ by $2$ .

$f' (x) = 6 x^{2} + \frac{d}{d x} (3 x^{2}) + \frac{d}{d x} (- 336 x)$

Step 1.1.1.3

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

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Step 1.1.1.3.1

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

$f' (x) = 6 x^{2} + 3 \frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 336 x)$

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 $n = 2$ .

$f' (x) = 6 x^{2} + 3 (2 x) + \frac{d}{d x} (- 336 x)$

Step 1.1.1.3.3

Multiply $2$ by $3$ .

$f' (x) = 6 x^{2} + 6 x + \frac{d}{d x} (- 336 x)$

Step 1.1.1.4

Evaluate $\frac{d}{d x} [- 336 x]$ .

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Step 1.1.1.4.1

Since $- 336$ is constant with respect to $x$ , the derivative of $- 336 x$ with respect to $x$ is $- 336 \frac{d}{d x} [x]$ .

$f' (x) = 6 x^{2} + 6 x - 336 \frac{d}{d x} x$

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 $n = 1$ .

$f' (x) = 6 x^{2} + 6 x - 336 \cdot 1$

Step 1.1.1.4.3

Multiply $- 336$ by $1$ .

$f' (x) = 6 x^{2} + 6 x - 336$

Step 1.1.2

Find the second derivative.

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Step 1.1.2.1

By the Sum Rule, the derivative of $6 x^{2} + 6 x - 336$ with respect to $x$ is $\frac{d}{d x} [6 x^{2}] + \frac{d}{d x} [6 x] + \frac{d}{d x} [- 336]$ .

$f'' (x) = \frac{d}{d x} (6 x^{2}) + \frac{d}{d x} (6 x) + \frac{d}{d x} (- 336)$

Step 1.1.2.2

Evaluate $\frac{d}{d x} [6 x^{2}]$ .

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Step 1.1.2.2.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x^{2}$ with respect to $x$ is $6 \frac{d}{d x} [x^{2}]$ .

$f'' (x) = 6 \frac{d}{d x} (x^{2}) + \frac{d}{d x} (6 x) + \frac{d}{d x} (- 336)$

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 $n = 2$ .

$f'' (x) = 6 (2 x) + \frac{d}{d x} (6 x) + \frac{d}{d x} (- 336)$

Step 1.1.2.2.3

Multiply $2$ by $6$ .

$f'' (x) = 12 x + \frac{d}{d x} (6 x) + \frac{d}{d x} (- 336)$

Step 1.1.2.3

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

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Step 1.1.2.3.1

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

$f'' (x) = 12 x + 6 \frac{d}{d x} (x) + \frac{d}{d x} (- 336)$

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 $n = 1$ .

$f'' (x) = 12 x + 6 \cdot 1 + \frac{d}{d x} (- 336)$

Step 1.1.2.3.3

Multiply $6$ by $1$ .

$f'' (x) = 12 x + 6 + \frac{d}{d x} (- 336)$

Step 1.1.2.4

Differentiate using the Constant Rule.

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Step 1.1.2.4.1

Since $- 336$ is constant with respect to $x$ , the derivative of $- 336$ with respect to $x$ is $0$ .

$f'' (x) = 12 x + 6 + 0$

Step 1.1.2.4.2

Add $12 x + 6$ and $0$ .

$f'' (x) = 12 x + 6$

Step 1.1.3

The second derivative of $f (x)$ with respect to $x$ is $12 x + 6$ .

$12 x + 6$

Step 1.2

Set the second derivative equal to $0$ then solve the equation $12 x + 6 = 0$ .

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Step 1.2.1

Set the second derivative equal to $0$ .

$12 x + 6 = 0$

Step 1.2.2

Subtract $6$ from both sides of the equation.

$12 x = - 6$

Step 1.2.3

Divide each term in $12 x = - 6$ by $12$ and simplify.

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Step 1.2.3.1

Divide each term in $12 x = - 6$ by $12$ .

$\frac{12 x}{12} = \frac{- 6}{12}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Cancel the common factor of $12$ .

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Step 1.2.3.2.1.1

Cancel the common factor.

$\frac{12 x}{12} = \frac{- 6}{12}$

Step 1.2.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 6}{12}$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Cancel the common factor of $- 6$ and $12$ .

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Step 1.2.3.3.1.1

Factor $6$ out of $- 6$ .

$x = \frac{6 (- 1)}{12}$

Step 1.2.3.3.1.2

Cancel the common factors.

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Step 1.2.3.3.1.2.1

Factor $6$ out of $12$ .

$x = \frac{6 \cdot - 1}{6 \cdot 2}$

Step 1.2.3.3.1.2.2

Cancel the common factor.

$x = \frac{6 \cdot - 1}{6 \cdot 2}$

Step 1.2.3.3.1.2.3

Rewrite the expression.

$x = \frac{- 1}{2}$

Step 1.2.3.3.2

Move the negative in front of the fraction.

$x = - \frac{1}{2}$

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.

$(- \infty, - \frac{1}{2}) \cup (- \frac{1}{2}, \infty)$

Step 4

Substitute any number from the interval $(- \infty, - \frac{1}{2})$ into the second derivative and evaluate to determine the concavity.

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Step 4.1

Replace the variable $x$ with $- 3$ in the expression.

$f'' (- 3) = 12 (- 3) + 6$

Step 4.2

Simplify the result.

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Step 4.2.1

Multiply $12$ by $- 3$ .

$f'' (- 3) = - 36 + 6$

Step 4.2.2

Add $- 36$ and $6$ .

$f'' (- 3) = - 30$

Step 4.2.3

The final answer is $- 30$ .

$- 30$

Step 4.3

The graph is concave down on the interval $(- \infty, - \frac{1}{2})$ because $f'' (- 3)$ is negative.

Concave down on $(- \infty, - \frac{1}{2})$ since $f'' (x)$ is negative

Step 5

Substitute any number from the interval $(- \frac{1}{2}, \infty)$ 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.

$f'' (0) = 12 (0) + 6$

Step 5.2

Simplify the result.

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Step 5.2.1

Multiply $12$ by $0$ .

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

Step 5.2.2

Add $0$ and $6$ .

$f'' (0) = 6$

Step 5.2.3

The final answer is $6$ .

$6$

Step 5.3

The graph is concave up on the interval $(- \frac{1}{2}, \infty)$ because $f'' (0)$ is positive.

Concave up on $(- \frac{1}{2}, \infty)$ 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 $(- \infty, - \frac{1}{2})$ since $f'' (x)$ is negative

Concave up on $(- \frac{1}{2}, \infty)$ since $f'' (x)$ is positive

Step 7