Calculus Examples

$f (x) = x^{4} - 6$

Step 1

Find the $x$ values where the second derivative is equal to $0$ .

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Find the second derivative.

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Find the first derivative.

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By the Sum Rule, the derivative of $x^{4} - 6$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 6]$ .

$f' (x) = \frac{d}{d x} (x^{4}) + \frac{d}{d x} (- 6)$

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

$f' (x) = 4 x^{3} + \frac{d}{d x} (- 6)$

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6$ with respect to $x$ is $0$ .

$f' (x) = 4 x^{3} + 0$

Add $4 x^{3}$ and $0$ .

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

Find the second derivative.

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

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

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

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

Multiply $3$ by $4$ .

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

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

$12 x^{2}$

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

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Set the second derivative equal to $0$ .

$12 x^{2} = 0$

Divide each term in $12 x^{2} = 0$ by $12$ and simplify.

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Divide each term in $12 x^{2} = 0$ by $12$ .

$\frac{12 x^{2}}{12} = \frac{0}{12}$

Simplify the left side.

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Cancel the common factor of $12$ .

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Cancel the common factor.

$\frac{12 x^{2}}{12} = \frac{0}{12}$

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

$x^{2} = \frac{0}{12}$

Simplify the right side.

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Divide $0$ by $12$ .

$x^{2} = 0$

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{0}$

Simplify $\pm \sqrt{0}$ .

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Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Plus or minus $0$ is $0$ .

$x = 0$

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.

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

Step 4

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

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

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

Simplify the result.

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

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

Multiply $12$ by $4$ .

$f'' (- 2) = 48$

The final answer is $48$ .

$48$

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 $(0, \infty)$ into the second derivative and evaluate to determine the concavity.

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

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

Simplify the result.

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

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

Multiply $12$ by $4$ .

$f'' (2) = 48$

The final answer is $48$ .

$48$

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 6

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

Step 7

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