Calculus Examples

$f (x) = x^{4} - 24 x^{2} + 12 x$

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

Differentiate.

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Step 1.1.1.1.1

By the Sum Rule, the derivative of $x^{4} - 24 x^{2} + 12 x$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 24 x^{2}] + \frac{d}{d x} [12 x]$ .

$f' (x) = \frac{d}{d x} (x^{4}) + \frac{d}{d x} (- 24 x^{2}) + \frac{d}{d x} (12 x)$

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 $n = 4$ .

$f' (x) = 4 x^{3} + \frac{d}{d x} (- 24 x^{2}) + \frac{d}{d x} (12 x)$

Step 1.1.1.2

Evaluate $\frac{d}{d x} [- 24 x^{2}]$ .

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Step 1.1.1.2.1

Since $- 24$ is constant with respect to $x$ , the derivative of $- 24 x^{2}$ with respect to $x$ is $- 24 \frac{d}{d x} [x^{2}]$ .

$f' (x) = 4 x^{3} - 24 \frac{d}{d x} x^{2} + \frac{d}{d x} (12 x)$

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 $n = 2$ .

$f' (x) = 4 x^{3} - 24 (2 x) + \frac{d}{d x} (12 x)$

Step 1.1.1.2.3

Multiply $2$ by $- 24$ .

$f' (x) = 4 x^{3} - 48 x + \frac{d}{d x} (12 x)$

Step 1.1.1.3

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

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Step 1.1.1.3.1

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

$f' (x) = 4 x^{3} - 48 x + 12 \frac{d}{d x} (x)$

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 $n = 1$ .

$f' (x) = 4 x^{3} - 48 x + 12 \cdot 1$

Step 1.1.1.3.3

Multiply $12$ by $1$ .

$f' (x) = 4 x^{3} - 48 x + 12$

Step 1.1.2

Find the second derivative.

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Step 1.1.2.1

By the Sum Rule, the derivative of $4 x^{3} - 48 x + 12$ with respect to $x$ is $\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- 48 x] + \frac{d}{d x} [12]$ .

$f'' (x) = \frac{d}{d x} (4 x^{3}) + \frac{d}{d x} (- 48 x) + \frac{d}{d x} (12)$

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

$f'' (x) = 4 \frac{d}{d x} (x^{3}) + \frac{d}{d x} (- 48 x) + \frac{d}{d x} (12)$

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 $n = 3$ .

$f'' (x) = 4 (3 x^{2}) + \frac{d}{d x} (- 48 x) + \frac{d}{d x} (12)$

Step 1.1.2.2.3

Multiply $3$ by $4$ .

$f'' (x) = 12 x^{2} + \frac{d}{d x} (- 48 x) + \frac{d}{d x} (12)$

Step 1.1.2.3

Evaluate $\frac{d}{d x} [- 48 x]$ .

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Step 1.1.2.3.1

Since $- 48$ is constant with respect to $x$ , the derivative of $- 48 x$ with respect to $x$ is $- 48 \frac{d}{d x} [x]$ .

$f'' (x) = 12 x^{2} - 48 \frac{d}{d x} x + \frac{d}{d x} (12)$

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 $n = 1$ .

$f'' (x) = 12 x^{2} - 48 \cdot 1 + \frac{d}{d x} (12)$

Step 1.1.2.3.3

Multiply $- 48$ by $1$ .

$f'' (x) = 12 x^{2} - 48 + \frac{d}{d x} (12)$

Step 1.1.2.4

Differentiate using the Constant Rule.

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Step 1.1.2.4.1

Since $12$ is constant with respect to $x$ , the derivative of $12$ with respect to $x$ is $0$ .

$f'' (x) = 12 x^{2} - 48 + 0$

Step 1.1.2.4.2

Add $12 x^{2} - 48$ and $0$ .

$f'' (x) = 12 x^{2} - 48$

Step 1.1.3

The second derivative of $f (x)$ with respect to $x$ is $12 x^{2} - 48$ .

$12 x^{2} - 48$

Step 1.2

Set the second derivative equal to $0$ then solve the equation $12 x^{2} - 48 = 0$ .

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Step 1.2.1

Set the second derivative equal to $0$ .

$12 x^{2} - 48 = 0$

Step 1.2.2

Add $48$ to both sides of the equation.

$12 x^{2} = 48$

Step 1.2.3

Divide each term in $12 x^{2} = 48$ by $12$ and simplify.

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Step 1.2.3.1

Divide each term in $12 x^{2} = 48$ by $12$ .

$\frac{12 x^{2}}{12} = \frac{48}{12}$

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.

$\frac{12 x^{2}}{12} = \frac{48}{12}$

Step 1.2.3.2.1.2

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

$x^{2} = \frac{48}{12}$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Divide $48$ by $12$ .

$x^{2} = 4$

Step 1.2.4

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

$x = \pm \sqrt{4}$

Step 1.2.5

Simplify $\pm \sqrt{4}$ .

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Step 1.2.5.1

Rewrite $4$ as $2^{2}$ .

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

Step 1.2.5.2

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

$x = \pm 2$

Step 1.2.6

The complete solution is the result of both the positive and negative portions of the solution.

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Step 1.2.6.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 2$

Step 1.2.6.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 2$

Step 1.2.6.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 2, - 2$

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.

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

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.

$f'' (- 4) = 12 {(- 4)}^{2} - 48$

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 $2$ .

$f'' (- 4) = 12 \cdot 16 - 48$

Step 4.2.1.2

Multiply $12$ by $16$ .

$f'' (- 4) = 192 - 48$

Step 4.2.2

Subtract $48$ from $192$ .

$f'' (- 4) = 144$

Step 4.2.3

The final answer is $144$ .

$144$

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 $(- 2, 2)$ 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.

$f'' (0) = 12 {(0)}^{2} - 48$

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

Raising $0$ to any positive power yields $0$ .

$f'' (0) = 12 \cdot 0 - 48$

Step 5.2.1.2

Multiply $12$ by $0$ .

$f'' (0) = 0 - 48$

Step 5.2.2

Subtract $48$ from $0$ .

$f'' (0) = - 48$

Step 5.2.3

The final answer is $- 48$ .

$- 48$

Step 5.3

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

Concave down on $(- 2, 2)$ 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.

$f'' (4) = 12 {(4)}^{2} - 48$

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 $2$ .

$f'' (4) = 12 \cdot 16 - 48$

Step 6.2.1.2

Multiply $12$ by $16$ .

$f'' (4) = 192 - 48$

Step 6.2.2

Subtract $48$ from $192$ .

$f'' (4) = 144$

Step 6.2.3

The final answer is $144$ .

$144$

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

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

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

Step 8