Calculus Examples

$f (x) = x^{4} - 12 x^{2} + 36$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Differentiate.

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Step 1.1.1.1

By the Sum Rule, the derivative of $x^{4} - 12 x^{2} + 36$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 12 x^{2}] + \frac{d}{d x} [36]$ .

$\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 12 x^{2}] + \frac{d}{d x} [36]$

Step 1.1.1.2

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

$4 x^{3} + \frac{d}{d x} [- 12 x^{2}] + \frac{d}{d x} [36]$

Step 1.1.2

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

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Step 1.1.2.1

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

$4 x^{3} - 12 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [36]$

Step 1.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$4 x^{3} - 12 (2 x) + \frac{d}{d x} [36]$

Step 1.1.2.3

Multiply $2$ by $- 12$ .

$4 x^{3} - 24 x + \frac{d}{d x} [36]$

Step 1.1.3

Differentiate using the Constant Rule.

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Step 1.1.3.1

Since $36$ is constant with respect to $x$ , the derivative of $36$ with respect to $x$ is $0$ .

$4 x^{3} - 24 x + 0$

Step 1.1.3.2

Add $4 x^{3} - 24 x$ and $0$ .

$f' (x) = 4 x^{3} - 24 x$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $4 x^{3} - 24 x$ .

$4 x^{3} - 24 x$

Step 2

Set the first derivative equal to $0$ then solve the equation $4 x^{3} - 24 x = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$4 x^{3} - 24 x = 0$

Step 2.2

Factor $4 x$ out of $4 x^{3} - 24 x$ .

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Step 2.2.1

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

$4 x (x^{2}) - 24 x = 0$

Step 2.2.2

Factor $4 x$ out of $- 24 x$ .

$4 x (x^{2}) + 4 x (- 6) = 0$

Step 2.2.3

Factor $4 x$ out of $4 x (x^{2}) + 4 x (- 6)$ .

$4 x (x^{2} - 6) = 0$

Step 2.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$x^{2} - 6 = 0$

Step 2.4

Set $x$ equal to $0$ .

$x = 0$

Step 2.5

Set $x^{2} - 6$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $x^{2} - 6$ equal to $0$ .

$x^{2} - 6 = 0$

Step 2.5.2

Solve $x^{2} - 6 = 0$ for $x$ .

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Step 2.5.2.1

Add $6$ to both sides of the equation.

$x^{2} = 6$

Step 2.5.2.2

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

$x = \pm \sqrt{6}$

Step 2.5.2.3

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

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Step 2.5.2.3.1

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

$x = \sqrt{6}$

Step 2.5.2.3.2

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

$x = - \sqrt{6}$

Step 2.5.2.3.3

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

$x = \sqrt{6}, - \sqrt{6}$

Step 2.6

The final solution is all the values that make $4 x (x^{2} - 6) = 0$ true.

$x = 0, \sqrt{6}, - \sqrt{6}$

Step 3

The values which make the derivative equal to $0$ are $0, \sqrt{6}, - \sqrt{6}$ .

$0, \sqrt{6}, - \sqrt{6}$

Step 4

Split $(- \infty, \infty)$ into separate intervals around the $x$ values that make the derivative $0$ or undefined.

$(- \infty, - \sqrt{6}) \cup (- \sqrt{6}, 0) \cup (0, \sqrt{6}) \cup (\sqrt{6}, \infty)$

Step 5

Substitute a value from the interval $(- \infty, - \sqrt{6})$ into the derivative to determine if the function is increasing or decreasing.

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Step 5.1

Replace the variable $x$ with $- 3.4494898$ in the expression.

$f' (- 3.4494898) = 4 {(- 3.4494898)}^{3} - 24 \cdot - 3.4494898$

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 $- 3.4494898$ to the power of $3$ .

$f' (- 3.4494898) = 4 \cdot - 41.04540972 - 24 \cdot - 3.4494898$

Step 5.2.1.2

Multiply $4$ by $- 41.04540972$ .

$f' (- 3.4494898) = - 164.18163891 - 24 \cdot - 3.4494898$

Step 5.2.1.3

Multiply $- 24$ by $- 3.4494898$ .

$f' (- 3.4494898) = - 164.18163891 + 82.7877552$

Step 5.2.2

Add $- 164.18163891$ and $82.7877552$ .

$f' (- 3.4494898) = - 81.39388371$

Step 5.2.3

The final answer is $- 81.39388371$ .

$- 81.39388371$

Step 5.3

At $x = - 3.4494898$ the derivative is $- 81.39388371$ . Since this is negative, the function is decreasing on $(- \infty, - \sqrt{6})$ .

Decreasing on $(- \infty, - \sqrt{6})$ since $f' (x) < 0$

Step 6

Substitute a value from the interval $(- 2.4494898, 0)$ into the derivative to determine if the function is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $- 1.2247449$ in the expression.

$f' (- 1.2247449) = 4 {(- 1.2247449)}^{3} - 24 \cdot - 1.2247449$

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 $- 1.2247449$ to the power of $3$ .

$f' (- 1.2247449) = 4 \cdot - 1.83711743 - 24 \cdot - 1.2247449$

Step 6.2.1.2

Multiply $4$ by $- 1.83711743$ .

$f' (- 1.2247449) = - 7.34846974 - 24 \cdot - 1.2247449$

Step 6.2.1.3

Multiply $- 24$ by $- 1.2247449$ .

$f' (- 1.2247449) = - 7.34846974 + 29.3938776$

Step 6.2.2

Add $- 7.34846974$ and $29.3938776$ .

$f' (- 1.2247449) = 22.04540785$

Step 6.2.3

The final answer is $22.04540785$ .

$22.04540785$

Step 6.3

At $x = - 1.2247449$ the derivative is $22.04540785$ . Since this is positive, the function is increasing on $(- 2.4494898, 0)$ .

Increasing on $(- \sqrt{6}, 0)$ since $f' (x) > 0$

Step 7

Substitute a value from the interval $(0, \sqrt{6})$ into the derivative to determine if the function is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $1.2247449$ in the expression.

$f' (1.2247449) = 4 {(1.2247449)}^{3} - 24 \cdot 1.2247449$

Step 7.2

Simplify the result.

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Step 7.2.1

Simplify each term.

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Step 7.2.1.1

Raise $1.2247449$ to the power of $3$ .

$f' (1.2247449) = 4 \cdot 1.83711743 - 24 \cdot 1.2247449$

Step 7.2.1.2

Multiply $4$ by $1.83711743$ .

$f' (1.2247449) = 7.34846974 - 24 \cdot 1.2247449$

Step 7.2.1.3

Multiply $- 24$ by $1.2247449$ .

$f' (1.2247449) = 7.34846974 - 29.3938776$

Step 7.2.2

Subtract $29.3938776$ from $7.34846974$ .

$f' (1.2247449) = - 22.04540785$

Step 7.2.3

The final answer is $- 22.04540785$ .

$- 22.04540785$

Step 7.3

At $x = 1.2247449$ the derivative is $- 22.04540785$ . Since this is negative, the function is decreasing on $(0, \sqrt{6})$ .

Decreasing on $(0, \sqrt{6})$ since $f' (x) < 0$

Step 8

Substitute a value from the interval $(\sqrt{6}, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Step 8.1

Replace the variable $x$ with $3.4494898$ in the expression.

$f' (3.4494898) = 4 {(3.4494898)}^{3} - 24 \cdot 3.4494898$

Step 8.2

Simplify the result.

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Step 8.2.1

Simplify each term.

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Step 8.2.1.1

Raise $3.4494898$ to the power of $3$ .

$f' (3.4494898) = 4 \cdot 41.04540972 - 24 \cdot 3.4494898$

Step 8.2.1.2

Multiply $4$ by $41.04540972$ .

$f' (3.4494898) = 164.18163891 - 24 \cdot 3.4494898$

Step 8.2.1.3

Multiply $- 24$ by $3.4494898$ .

$f' (3.4494898) = 164.18163891 - 82.7877552$

Step 8.2.2

Subtract $82.7877552$ from $164.18163891$ .

$f' (3.4494898) = 81.39388371$

Step 8.2.3

The final answer is $81.39388371$ .

$81.39388371$

Step 8.3

At $x = 3.4494898$ the derivative is $81.39388371$ . Since this is positive, the function is increasing on $(\sqrt{6}, \infty)$ .

Increasing on $(\sqrt{6}, \infty)$ since $f' (x) > 0$

Step 9

List the intervals on which the function is increasing and decreasing.

Increasing on: $(- \sqrt{6}, 0), (\sqrt{6}, \infty)$

Decreasing on: $(- \infty, - \sqrt{6}), (0, \sqrt{6})$

Step 10

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