Calculus Examples

$f (x) = x^{4} - 32 x^{2}$

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 $x^{4} - 32 x^{2}$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 32 x^{2}]$ .

$\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 32 x^{2}]$

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 $n = 4$ .

$4 x^{3} + \frac{d}{d x} [- 32 x^{2}]$

Step 1.1.2

Evaluate $\frac{d}{d x} [- 32 x^{2}]$ .

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Step 1.1.2.1

Since $- 32$ is constant with respect to $x$ , the derivative of $- 32 x^{2}$ with respect to $x$ is $- 32 \frac{d}{d x} [x^{2}]$ .

$4 x^{3} - 32 \frac{d}{d x} [x^{2}]$

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 $n = 2$ .

$4 x^{3} - 32 (2 x)$

Step 1.1.2.3

Multiply $2$ by $- 32$ .

$f' (x) = 4 x^{3} - 64 x$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $4 x^{3} - 64 x$ .

$4 x^{3} - 64 x$

Step 2

Set the first derivative equal to $0$ then solve the equation $4 x^{3} - 64 x = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$4 x^{3} - 64 x = 0$

Step 2.2

Factor the left side of the equation.

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Step 2.2.1

Factor $4 x$ out of $4 x^{3} - 64 x$ .

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Step 2.2.1.1

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

$4 x (x^{2}) - 64 x = 0$

Step 2.2.1.2

Factor $4 x$ out of $- 64 x$ .

$4 x (x^{2}) + 4 x (- 16) = 0$

Step 2.2.1.3

Factor $4 x$ out of $4 x (x^{2}) + 4 x (- 16)$ .

$4 x (x^{2} - 16) = 0$

Step 2.2.2

Rewrite $16$ as $4^{2}$ .

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

Step 2.2.3

Factor.

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Step 2.2.3.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 4$ .

$4 x ((x + 4) (x - 4)) = 0$

Step 2.2.3.2

Remove unnecessary parentheses.

$4 x (x + 4) (x - 4) = 0$

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 $0$ .

$x = 0$

$x + 4 = 0$

$x - 4 = 0$

Step 2.4

Set $x$ equal to $0$ .

$x = 0$

Step 2.5

Set $x + 4$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Step 2.5.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 2.6

Set $x - 4$ equal to $0$ and solve for $x$ .

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Step 2.6.1

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 2.6.2

Add $4$ to both sides of the equation.

$x = 4$

Step 2.7

The final solution is all the values that make $4 x (x + 4) (x - 4) = 0$ true.

$x = 0, - 4, 4$

Step 3

The values which make the derivative equal to $0$ are $0, - 4, 4$ .

$0, - 4, 4$

Step 4

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

$(- \infty, - 4) \cup (- 4, 0) \cup (0, 4) \cup (4, \infty)$

Step 5

Substitute a value from the interval $(- \infty, - 4)$ into the derivative to determine if the function is increasing or decreasing.

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Step 5.1

Replace the variable $x$ with $- 5$ in the expression.

$f' (- 5) = 4 {(- 5)}^{3} - 64 \cdot - 5$

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 $- 5$ to the power of $3$ .

$f' (- 5) = 4 \cdot - 125 - 64 \cdot - 5$

Step 5.2.1.2

Multiply $4$ by $- 125$ .

$f' (- 5) = - 500 - 64 \cdot - 5$

Step 5.2.1.3

Multiply $- 64$ by $- 5$ .

$f' (- 5) = - 500 + 320$

Step 5.2.2

Add $- 500$ and $320$ .

$f' (- 5) = - 180$

Step 5.2.3

The final answer is $- 180$ .

$- 180$

Step 5.3

At $x = - 5$ the derivative is $- 180$ . Since this is negative, the function is decreasing on $(- \infty, - 4)$ .

Decreasing on $(- \infty, - 4)$ since $f' (x) < 0$

Step 6

Substitute a value from the interval $(- 4, 0)$ into the derivative to determine if the function is increasing or decreasing.

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Step 6.1

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

$f' (- 2) = 4 {(- 2)}^{3} - 64 \cdot - 2$

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 $3$ .

$f' (- 2) = 4 \cdot - 8 - 64 \cdot - 2$

Step 6.2.1.2

Multiply $4$ by $- 8$ .

$f' (- 2) = - 32 - 64 \cdot - 2$

Step 6.2.1.3

Multiply $- 64$ by $- 2$ .

$f' (- 2) = - 32 + 128$

Step 6.2.2

Add $- 32$ and $128$ .

$f' (- 2) = 96$

Step 6.2.3

The final answer is $96$ .

$96$

Step 6.3

At $x = - 2$ the derivative is $96$ . Since this is positive, the function is increasing on $(- 4, 0)$ .

Increasing on $(- 4, 0)$ since $f' (x) > 0$

Step 7

Substitute a value from the interval $(0, 4)$ into the derivative to determine if the function is increasing or decreasing.

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Step 7.1

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

$f' (2) = 4 {(2)}^{3} - 64 \cdot 2$

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 $2$ to the power of $3$ .

$f' (2) = 4 \cdot 8 - 64 \cdot 2$

Step 7.2.1.2

Multiply $4$ by $8$ .

$f' (2) = 32 - 64 \cdot 2$

Step 7.2.1.3

Multiply $- 64$ by $2$ .

$f' (2) = 32 - 128$

Step 7.2.2

Subtract $128$ from $32$ .

$f' (2) = - 96$

Step 7.2.3

The final answer is $- 96$ .

$- 96$

Step 7.3

At $x = 2$ the derivative is $- 96$ . Since this is negative, the function is decreasing on $(0, 4)$ .

Decreasing on $(0, 4)$ since $f' (x) < 0$

Step 8

Substitute a value from the interval $(4, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Step 8.1

Replace the variable $x$ with $5$ in the expression.

$f' (5) = 4 {(5)}^{3} - 64 \cdot 5$

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 $5$ to the power of $3$ .

$f' (5) = 4 \cdot 125 - 64 \cdot 5$

Step 8.2.1.2

Multiply $4$ by $125$ .

$f' (5) = 500 - 64 \cdot 5$

Step 8.2.1.3

Multiply $- 64$ by $5$ .

$f' (5) = 500 - 320$

Step 8.2.2

Subtract $320$ from $500$ .

$f' (5) = 180$

Step 8.2.3

The final answer is $180$ .

$180$

Step 8.3

At $x = 5$ the derivative is $180$ . Since this is positive, the function is increasing on $(4, \infty)$ .

Increasing on $(4, \infty)$ since $f' (x) > 0$

Step 9

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

Increasing on: $(- 4, 0), (4, \infty)$

Decreasing on: $(- \infty, - 4), (0, 4)$

Step 10