Calculus Examples

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

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

$f' (x) = \frac{d}{d x} (x^{4}) + \frac{d}{d x} (- 32 x) + \frac{d}{d x} (4)$

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 $n = 4$ .

$f' (x) = 4 x^{3} + \frac{d}{d x} (- 32 x) + \frac{d}{d x} (4)$

Step 1.1.2

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

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Step 1.1.2.1

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

$f' (x) = 4 x^{3} - 32 \frac{d}{d x} x + \frac{d}{d x} (4)$

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 $n = 1$ .

$f' (x) = 4 x^{3} - 32 \cdot 1 + \frac{d}{d x} (4)$

Step 1.1.2.3

Multiply $- 32$ by $1$ .

$f' (x) = 4 x^{3} - 32 + \frac{d}{d x} (4)$

Step 1.1.3

Differentiate using the Constant Rule.

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Step 1.1.3.1

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$f' (x) = 4 x^{3} - 32 + 0$

Step 1.1.3.2

Add $4 x^{3} - 32$ and $0$ .

$f' (x) = 4 x^{3} - 32$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $4 x^{3} - 32$ .

$4 x^{3} - 32$

Step 2

Set the first derivative equal to $0$ then solve the equation $4 x^{3} - 32 = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$4 x^{3} - 32 = 0$

Step 2.2

Add $32$ to both sides of the equation.

$4 x^{3} = 32$

Step 2.3

Subtract $32$ from both sides of the equation.

$4 x^{3} - 32 = 0$

Step 2.4

Factor the left side of the equation.

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Step 2.4.1

Factor $4$ out of $4 x^{3} - 32$ .

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Step 2.4.1.1

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

$4 (x^{3}) - 32 = 0$

Step 2.4.1.2

Factor $4$ out of $- 32$ .

$4 x^{3} + 4 \cdot - 8 = 0$

Step 2.4.1.3

Factor $4$ out of $4 x^{3} + 4 \cdot - 8$ .

$4 (x^{3} - 8) = 0$

Step 2.4.2

Rewrite $8$ as $2^{3}$ .

$4 (x^{3} - 2^{3}) = 0$

Step 2.4.3

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = x$ and $b = 2$ .

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

Step 2.4.4

Factor.

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Step 2.4.4.1

Simplify.

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Step 2.4.4.1.1

Move $2$ to the left of $x$ .

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

Step 2.4.4.1.2

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

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

Step 2.4.4.2

Remove unnecessary parentheses.

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

Step 2.5

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

$x - 2 = 0$

$x^{2} + 2 x + 4 = 0$

Step 2.6

Set $x - 2$ equal to $0$ and solve for $x$ .

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Step 2.6.1

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 2.6.2

Add $2$ to both sides of the equation.

$x = 2$

Step 2.7

Set $x^{2} + 2 x + 4$ equal to $0$ and solve for $x$ .

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Step 2.7.1

Set $x^{2} + 2 x + 4$ equal to $0$ .

$x^{2} + 2 x + 4 = 0$

Step 2.7.2

Solve $x^{2} + 2 x + 4 = 0$ for $x$ .

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Step 2.7.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 2.7.2.2

Substitute the values $a = 1$ , $b = 2$ , and $c = 4$ into the quadratic formula and solve for $x$ .

$\frac{- 2 \pm \sqrt{2^{2} - 4 \cdot (1 \cdot 4)}}{2 \cdot 1}$

Step 2.7.2.3

Simplify.

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Step 2.7.2.3.1

Simplify the numerator.

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Step 2.7.2.3.1.1

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

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 1 \cdot 4}}{2 \cdot 1}$

Step 2.7.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 4$ .

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Step 2.7.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 4}}{2 \cdot 1}$

Step 2.7.2.3.1.2.2

Multiply $- 4$ by $4$ .

$x = \frac{- 2 \pm \sqrt{4 - 16}}{2 \cdot 1}$

Step 2.7.2.3.1.3

Subtract $16$ from $4$ .

$x = \frac{- 2 \pm \sqrt{- 12}}{2 \cdot 1}$

Step 2.7.2.3.1.4

Rewrite $- 12$ as $- 1 (12)$ .

$x = \frac{- 2 \pm \sqrt{- 1 \cdot 12}}{2 \cdot 1}$

Step 2.7.2.3.1.5

Rewrite $\sqrt{- 1 (12)}$ as $\sqrt{- 1} \cdot \sqrt{12}$ .

$x = \frac{- 2 \pm \sqrt{- 1} \cdot \sqrt{12}}{2 \cdot 1}$

Step 2.7.2.3.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 2 \pm i \cdot \sqrt{12}}{2 \cdot 1}$

Step 2.7.2.3.1.7

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 2.7.2.3.1.7.1

Factor $4$ out of $12$ .

$x = \frac{- 2 \pm i \cdot \sqrt{4 (3)}}{2 \cdot 1}$

Step 2.7.2.3.1.7.2

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

$x = \frac{- 2 \pm i \cdot \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

Step 2.7.2.3.1.8

Pull terms out from under the radical.

$x = \frac{- 2 \pm i \cdot (2 \sqrt{3})}{2 \cdot 1}$

Step 2.7.2.3.1.9

Move $2$ to the left of $i$ .

$x = \frac{- 2 \pm 2 i \sqrt{3}}{2 \cdot 1}$

Step 2.7.2.3.2

Multiply $2$ by $1$ .

$x = \frac{- 2 \pm 2 i \sqrt{3}}{2}$

Step 2.7.2.3.3

Simplify $\frac{- 2 \pm 2 i \sqrt{3}}{2}$ .

$x = - 1 \pm i \sqrt{3}$

Step 2.7.2.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 2.7.2.4.1

Simplify the numerator.

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Step 2.7.2.4.1.1

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

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 1 \cdot 4}}{2 \cdot 1}$

Step 2.7.2.4.1.2

Multiply $- 4 \cdot 1 \cdot 4$ .

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Step 2.7.2.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 4}}{2 \cdot 1}$

Step 2.7.2.4.1.2.2

Multiply $- 4$ by $4$ .

$x = \frac{- 2 \pm \sqrt{4 - 16}}{2 \cdot 1}$

Step 2.7.2.4.1.3

Subtract $16$ from $4$ .

$x = \frac{- 2 \pm \sqrt{- 12}}{2 \cdot 1}$

Step 2.7.2.4.1.4

Rewrite $- 12$ as $- 1 (12)$ .

$x = \frac{- 2 \pm \sqrt{- 1 \cdot 12}}{2 \cdot 1}$

Step 2.7.2.4.1.5

Rewrite $\sqrt{- 1 (12)}$ as $\sqrt{- 1} \cdot \sqrt{12}$ .

$x = \frac{- 2 \pm \sqrt{- 1} \cdot \sqrt{12}}{2 \cdot 1}$

Step 2.7.2.4.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 2 \pm i \cdot \sqrt{12}}{2 \cdot 1}$

Step 2.7.2.4.1.7

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 2.7.2.4.1.7.1

Factor $4$ out of $12$ .

$x = \frac{- 2 \pm i \cdot \sqrt{4 (3)}}{2 \cdot 1}$

Step 2.7.2.4.1.7.2

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

$x = \frac{- 2 \pm i \cdot \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

Step 2.7.2.4.1.8

Pull terms out from under the radical.

$x = \frac{- 2 \pm i \cdot (2 \sqrt{3})}{2 \cdot 1}$

Step 2.7.2.4.1.9

Move $2$ to the left of $i$ .

$x = \frac{- 2 \pm 2 i \sqrt{3}}{2 \cdot 1}$

Step 2.7.2.4.2

Multiply $2$ by $1$ .

$x = \frac{- 2 \pm 2 i \sqrt{3}}{2}$

Step 2.7.2.4.3

Simplify $\frac{- 2 \pm 2 i \sqrt{3}}{2}$ .

$x = - 1 \pm i \sqrt{3}$

Step 2.7.2.4.4

Change the $\pm$ to $+$ .

$x = - 1 + i \sqrt{3}$

Step 2.7.2.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 2.7.2.5.1

Simplify the numerator.

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Step 2.7.2.5.1.1

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

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 1 \cdot 4}}{2 \cdot 1}$

Step 2.7.2.5.1.2

Multiply $- 4 \cdot 1 \cdot 4$ .

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Step 2.7.2.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 4}}{2 \cdot 1}$

Step 2.7.2.5.1.2.2

Multiply $- 4$ by $4$ .

$x = \frac{- 2 \pm \sqrt{4 - 16}}{2 \cdot 1}$

Step 2.7.2.5.1.3

Subtract $16$ from $4$ .

$x = \frac{- 2 \pm \sqrt{- 12}}{2 \cdot 1}$

Step 2.7.2.5.1.4

Rewrite $- 12$ as $- 1 (12)$ .

$x = \frac{- 2 \pm \sqrt{- 1 \cdot 12}}{2 \cdot 1}$

Step 2.7.2.5.1.5

Rewrite $\sqrt{- 1 (12)}$ as $\sqrt{- 1} \cdot \sqrt{12}$ .

$x = \frac{- 2 \pm \sqrt{- 1} \cdot \sqrt{12}}{2 \cdot 1}$

Step 2.7.2.5.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 2 \pm i \cdot \sqrt{12}}{2 \cdot 1}$

Step 2.7.2.5.1.7

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 2.7.2.5.1.7.1

Factor $4$ out of $12$ .

$x = \frac{- 2 \pm i \cdot \sqrt{4 (3)}}{2 \cdot 1}$

Step 2.7.2.5.1.7.2

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

$x = \frac{- 2 \pm i \cdot \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

Step 2.7.2.5.1.8

Pull terms out from under the radical.

$x = \frac{- 2 \pm i \cdot (2 \sqrt{3})}{2 \cdot 1}$

Step 2.7.2.5.1.9

Move $2$ to the left of $i$ .

$x = \frac{- 2 \pm 2 i \sqrt{3}}{2 \cdot 1}$

Step 2.7.2.5.2

Multiply $2$ by $1$ .

$x = \frac{- 2 \pm 2 i \sqrt{3}}{2}$

Step 2.7.2.5.3

Simplify $\frac{- 2 \pm 2 i \sqrt{3}}{2}$ .

$x = - 1 \pm i \sqrt{3}$

Step 2.7.2.5.4

Change the $\pm$ to $-$ .

$x = - 1 - i \sqrt{3}$

Step 2.7.2.6

The final answer is the combination of both solutions.

$x = - 1 + i \sqrt{3}, - 1 - i \sqrt{3}$

Step 2.8

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

$x = 2, - 1 + i \sqrt{3}, - 1 - i \sqrt{3}$

Step 3

The values which make the derivative equal to $0$ are $2$ .

$2$

Step 4

After finding the point that makes the derivative $f' (x) = 4 x^{3} - 32$ equal to $0$ or undefined, the interval to check where $f (x) = x^{4} - 32 x + 4$ is increasing and where it is decreasing is $(- \infty, 2) \cup (2, \infty)$ .

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

Step 5

Substitute a value from the interval $(- \infty, 2)$ into the derivative to determine if the function is increasing or decreasing.

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Step 5.1

Replace the variable $x$ with $1$ in the expression.

$f' (1) = 4 {(1)}^{3} - 32$

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.

$f' (1) = 4 \cdot 1 - 32$

Step 5.2.1.2

Multiply $4$ by $1$ .

$f' (1) = 4 - 32$

Step 5.2.2

Subtract $32$ from $4$ .

$f' (1) = - 28$

Step 5.2.3

The final answer is $- 28$ .

$- 28$

Step 5.3

At $x = 1$ the derivative is $- 28$ . Since this is negative, the function is decreasing on $(- \infty, 2)$ .

Decreasing on $(- \infty, 2)$ since $f' (x) < 0$

Step 6

Substitute a value from the interval $(2, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $3$ in the expression.

$f' (3) = 4 {(3)}^{3} - 32$

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 $3$ to the power of $3$ .

$f' (3) = 4 \cdot 27 - 32$

Step 6.2.1.2

Multiply $4$ by $27$ .

$f' (3) = 108 - 32$

Step 6.2.2

Subtract $32$ from $108$ .

$f' (3) = 76$

Step 6.2.3

The final answer is $76$ .

$76$

Step 6.3

At $x = 3$ the derivative is $76$ . Since this is positive, the function is increasing on $(2, \infty)$ .

Increasing on $(2, \infty)$ since $f' (x) > 0$

Step 7

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

Increasing on: $(2, \infty)$

Decreasing on: $(- \infty, 2)$

Step 8