Calculus Examples

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

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 $\frac{1}{2} x^{4} + 2 x^{3}$ with respect to $x$ is $\frac{d}{d x} [\frac{1}{2} x^{4}] + \frac{d}{d x} [2 x^{3}]$ .

$f' (x) = \frac{d}{d x} (\frac{1}{2} x^{4}) + \frac{d}{d x} (2 x^{3})$

Step 1.1.1.2

Evaluate $\frac{d}{d x} [\frac{1}{2} x^{4}]$ .

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Step 1.1.1.2.1

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{1}{2} x^{4}$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [x^{4}]$ .

$f' (x) = \frac{1}{2} \cdot \frac{d}{d x} (x^{4}) + \frac{d}{d x} (2 x^{3})$

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 $n = 4$ .

$f' (x) = \frac{1}{2} \cdot (4 x^{3}) + \frac{d}{d x} (2 x^{3})$

Step 1.1.1.2.3

Combine $4$ and $\frac{1}{2}$ .

$f' (x) = \frac{4}{2} x^{3} + \frac{d}{d x} (2 x^{3})$

Step 1.1.1.2.4

Combine $\frac{4}{2}$ and $x^{3}$ .

$f' (x) = \frac{4 x^{3}}{2} + \frac{d}{d x} (2 x^{3})$

Step 1.1.1.2.5

Cancel the common factor of $4$ and $2$ .

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Step 1.1.1.2.5.1

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

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

Step 1.1.1.2.5.2

Cancel the common factors.

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Step 1.1.1.2.5.2.1

Factor $2$ out of $2$ .

$f' (x) = \frac{2 (2 x^{3})}{2 (1)} + \frac{d}{d x} (2 x^{3})$

Step 1.1.1.2.5.2.2

Cancel the common factor.

$f' (x) = \frac{2 (2 x^{3})}{2 \cdot 1} + \frac{d}{d x} (2 x^{3})$

Step 1.1.1.2.5.2.3

Rewrite the expression.

$f' (x) = \frac{2 x^{3}}{1} + \frac{d}{d x} (2 x^{3})$

Step 1.1.1.2.5.2.4

Divide $2 x^{3}$ by $1$ .

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

Step 1.1.1.3

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

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Step 1.1.1.3.1

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

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

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 $n = 3$ .

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

Step 1.1.1.3.3

Multiply $3$ by $2$ .

$f' (x) = 2 x^{3} + 6 x^{2}$

Step 1.1.2

Find the second derivative.

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Step 1.1.2.1

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

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

Step 1.1.2.2

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

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Step 1.1.2.2.1

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

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

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 $n = 3$ .

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

Step 1.1.2.2.3

Multiply $3$ by $2$ .

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

Step 1.1.2.3

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

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Step 1.1.2.3.1

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

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

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 $n = 2$ .

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

Step 1.1.2.3.3

Multiply $2$ by $6$ .

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

Step 1.1.3

The second derivative of $f (x)$ with respect to $x$ is $6 x^{2} + 12 x$ .

$6 x^{2} + 12 x$

Step 1.2

Set the second derivative equal to $0$ then solve the equation $6 x^{2} + 12 x = 0$ .

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Step 1.2.1

Set the second derivative equal to $0$ .

$6 x^{2} + 12 x = 0$

Step 1.2.2

Factor $6 x$ out of $6 x^{2} + 12 x$ .

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Step 1.2.2.1

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

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

Step 1.2.2.2

Factor $6 x$ out of $12 x$ .

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

Step 1.2.2.3

Factor $6 x$ out of $6 x (x) + 6 x (2)$ .

$6 x (x + 2) = 0$

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 $0$ .

$x = 0$

$x + 2 = 0$

Step 1.2.4

Set $x$ equal to $0$ .

$x = 0$

Step 1.2.5

Set $x + 2$ equal to $0$ and solve for $x$ .

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Step 1.2.5.1

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 1.2.5.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 1.2.6

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

$x = 0, - 2$

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.

$(- \infty, - 2) \cup (- 2, 0) \cup (0, \infty)$

Step 4

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

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Step 4.1

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

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

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

$f'' (- 4) = 6 \cdot 16 + 12 (- 4)$

Step 4.2.1.2

Multiply $6$ by $16$ .

$f'' (- 4) = 96 + 12 (- 4)$

Step 4.2.1.3

Multiply $12$ by $- 4$ .

$f'' (- 4) = 96 - 48$

Step 4.2.2

Subtract $48$ from $96$ .

$f'' (- 4) = 48$

Step 4.2.3

The final answer is $48$ .

$48$

Step 4.3

The graph is concave up on the interval $(- \infty, - 2)$ because $f'' (- 4)$ is positive.

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

Step 5

Substitute any number from the interval $(- 2, 0)$ into the second derivative and evaluate to determine the concavity.

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Step 5.1

Replace the variable $x$ with $- 1$ in the expression.

$f'' (- 1) = 6 {(- 1)}^{2} + 12 (- 1)$

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 $- 1$ to the power of $2$ .

$f'' (- 1) = 6 \cdot 1 + 12 (- 1)$

Step 5.2.1.2

Multiply $6$ by $1$ .

$f'' (- 1) = 6 + 12 (- 1)$

Step 5.2.1.3

Multiply $12$ by $- 1$ .

$f'' (- 1) = 6 - 12$

Step 5.2.2

Subtract $12$ from $6$ .

$f'' (- 1) = - 6$

Step 5.2.3

The final answer is $- 6$ .

$- 6$

Step 5.3

The graph is concave down on the interval $(- 2, 0)$ because $f'' (- 1)$ is negative.

Concave down on $(- 2, 0)$ since $f'' (x)$ is negative

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.

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

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 $2$ .

$f'' (2) = 6 \cdot 4 + 12 (2)$

Step 6.2.1.2

Multiply $6$ by $4$ .

$f'' (2) = 24 + 12 (2)$

Step 6.2.1.3

Multiply $12$ by $2$ .

$f'' (2) = 24 + 24$

Step 6.2.2

Add $24$ and $24$ .

$f'' (2) = 48$

Step 6.2.3

The final answer is $48$ .

$48$

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 up on $(- \infty, - 2)$ since $f'' (x)$ is positive

Concave down on $(- 2, 0)$ since $f'' (x)$ is negative

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

Step 8