Calculus Examples

$f (x) = x^{4} - 4 x^{3}$

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

Differentiate.

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Step 1.1.1.1.1

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

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

Step 1.1.1.1.2

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

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

Step 1.1.1.2

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

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Step 1.1.1.2.1

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

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

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 $n = 3$ .

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

Step 1.1.1.2.3

Multiply $3$ by $- 4$ .

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

Step 1.1.2

Find the second derivative.

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Step 1.1.2.1

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

$f'' (x) = \frac{d}{d x} (4 x^{3}) + \frac{d}{d x} (- 12 x^{2})$

Step 1.1.2.2

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

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Step 1.1.2.2.1

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

$f'' (x) = 4 \frac{d}{d x} (x^{3}) + \frac{d}{d x} (- 12 x^{2})$

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 $n = 3$ .

$f'' (x) = 4 (3 x^{2}) + \frac{d}{d x} (- 12 x^{2})$

Step 1.1.2.2.3

Multiply $3$ by $4$ .

$f'' (x) = 12 x^{2} + \frac{d}{d x} (- 12 x^{2})$

Step 1.1.2.3

Evaluate $\frac{d}{d x} [- 12 x^{2}]$ .

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Step 1.1.2.3.1

Since $- 12$ is constant with respect to $x$ , the derivative of $- 12 x^{2}$ with respect to $x$ is $- 12 \frac{d}{d x} [x^{2}]$ .

$f'' (x) = 12 x^{2} - 12 \frac{d}{d x} x^{2}$

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 $n = 2$ .

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

Step 1.1.2.3.3

Multiply $2$ by $- 12$ .

$f'' (x) = 12 x^{2} - 24 x$

Step 1.1.3

The second derivative of $f (x)$ with respect to $x$ is $12 x^{2} - 24 x$ .

$12 x^{2} - 24 x$

Step 1.2

Set the second derivative equal to $0$ then solve the equation $12 x^{2} - 24 x = 0$ .

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Step 1.2.1

Set the second derivative equal to $0$ .

$12 x^{2} - 24 x = 0$

Step 1.2.2

Factor $12 x$ out of $12 x^{2} - 24 x$ .

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Step 1.2.2.1

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

$12 x (x) - 24 x = 0$

Step 1.2.2.2

Factor $12 x$ out of $- 24 x$ .

$12 x (x) + 12 x (- 2) = 0$

Step 1.2.2.3

Factor $12 x$ out of $12 x (x) + 12 x (- 2)$ .

$12 x (x - 2) = 0$

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

Add $2$ to both sides of the equation.

$x = 2$

Step 1.2.6

The final solution is all the values that make $12 x (x - 2) = 0$ 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.

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

Step 4

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

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Step 4.1

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

$f'' (- 2) = 12 {(- 2)}^{2} - 24 \cdot - 2$

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

$f'' (- 2) = 12 \cdot 4 - 24 \cdot - 2$

Step 4.2.1.2

Multiply $12$ by $4$ .

$f'' (- 2) = 48 - 24 \cdot - 2$

Step 4.2.1.3

Multiply $- 24$ by $- 2$ .

$f'' (- 2) = 48 + 48$

Step 4.2.2

Add $48$ and $48$ .

$f'' (- 2) = 96$

Step 4.2.3

The final answer is $96$ .

$96$

Step 4.3

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

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

Step 5

Substitute any number from the interval $(0, 2)$ 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.

$f'' (1) = 12 {(1)}^{2} - 24 \cdot 1$

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

One to any power is one.

$f'' (1) = 12 \cdot 1 - 24 \cdot 1$

Step 5.2.1.2

Multiply $12$ by $1$ .

$f'' (1) = 12 - 24 \cdot 1$

Step 5.2.1.3

Multiply $- 24$ by $1$ .

$f'' (1) = 12 - 24$

Step 5.2.2

Subtract $24$ from $12$ .

$f'' (1) = - 12$

Step 5.2.3

The final answer is $- 12$ .

$- 12$

Step 5.3

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

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

Step 6

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

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Step 6.1

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

$f'' (4) = 12 {(4)}^{2} - 24 \cdot 4$

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 $4$ to the power of $2$ .

$f'' (4) = 12 \cdot 16 - 24 \cdot 4$

Step 6.2.1.2

Multiply $12$ by $16$ .

$f'' (4) = 192 - 24 \cdot 4$

Step 6.2.1.3

Multiply $- 24$ by $4$ .

$f'' (4) = 192 - 96$

Step 6.2.2

Subtract $96$ from $192$ .

$f'' (4) = 96$

Step 6.2.3

The final answer is $96$ .

$96$

Step 6.3

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

Concave up on $(2, \infty)$ 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 $(- \infty, 0)$ since $f'' (x)$ is positive

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

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

Step 8

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