Calculus Examples

$f (x) = \frac{1}{4} x^{4} - 8 x^{2}$

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

By the Sum Rule, the derivative of $\frac{1}{4} x^{4} - 8 x^{2}$ with respect to $x$ is $\frac{d}{d x} [\frac{1}{4} x^{4}] + \frac{d}{d x} [- 8 x^{2}]$ .

$\frac{d}{d x} [\frac{1}{4} x^{4}] + \frac{d}{d x} [- 8 x^{2}]$

Step 1.1.2

Evaluate $\frac{d}{d x} [\frac{1}{4} x^{4}]$ .

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Step 1.1.2.1

Since $\frac{1}{4}$ is constant with respect to $x$ , the derivative of $\frac{1}{4} x^{4}$ with respect to $x$ is $\frac{1}{4} \frac{d}{d x} [x^{4}]$ .

$\frac{1}{4} \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 8 x^{2}]$

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 $n = 4$ .

$\frac{1}{4} (4 x^{3}) + \frac{d}{d x} [- 8 x^{2}]$

Step 1.1.2.3

Combine $4$ and $\frac{1}{4}$ .

$\frac{4}{4} x^{3} + \frac{d}{d x} [- 8 x^{2}]$

Step 1.1.2.4

Combine $\frac{4}{4}$ and $x^{3}$ .

$\frac{4 x^{3}}{4} + \frac{d}{d x} [- 8 x^{2}]$

Step 1.1.2.5

Cancel the common factor of $4$ .

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Step 1.1.2.5.1

Cancel the common factor.

$\frac{4 x^{3}}{4} + \frac{d}{d x} [- 8 x^{2}]$

Step 1.1.2.5.2

Divide $x^{3}$ by $1$ .

$x^{3} + \frac{d}{d x} [- 8 x^{2}]$

Step 1.1.3

Evaluate $\frac{d}{d x} [- 8 x^{2}]$ .

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Step 1.1.3.1

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8 x^{2}$ with respect to $x$ is $- 8 \frac{d}{d x} [x^{2}]$ .

$x^{3} - 8 \frac{d}{d x} [x^{2}]$

Step 1.1.3.2

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

$x^{3} - 8 (2 x)$

Step 1.1.3.3

Multiply $2$ by $- 8$ .

$f' (x) = x^{3} - 16 x$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $x^{3} - 16 x$ .

$x^{3} - 16 x$

Step 2

Set the first derivative equal to $0$ then solve the equation $x^{3} - 16 x = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$x^{3} - 16 x = 0$

Step 2.2

Factor the left side of the equation.

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Step 2.2.1

Factor $x$ out of $x^{3} - 16 x$ .

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Step 2.2.1.1

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

$x \cdot x^{2} - 16 x = 0$

Step 2.2.1.2

Factor $x$ out of $- 16 x$ .

$x \cdot x^{2} + x \cdot - 16 = 0$

Step 2.2.1.3

Factor $x$ out of $x \cdot x^{2} + x \cdot - 16$ .

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

Step 2.2.2

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

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

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 $b = 4$ .

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

Step 2.2.3.2

Remove unnecessary parentheses.

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

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

$x = 0, - 4, 4$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

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.

Step 4

Evaluate $\frac{1}{4} x^{4} - 8 x^{2}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = 0$ .

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Step 4.1.1

Substitute $0$ for $x$ .

$\frac{1}{4} \cdot {(0)}^{4} - 8 {(0)}^{2}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Simplify each term.

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Step 4.1.2.1.1

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

$\frac{1}{4} \cdot 0 - 8 {(0)}^{2}$

Step 4.1.2.1.2

Multiply $\frac{1}{4}$ by $0$ .

$0 - 8 {(0)}^{2}$

Step 4.1.2.1.3

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

$0 - 8 \cdot 0$

Step 4.1.2.1.4

Multiply $- 8$ by $0$ .

$0 + 0$

Step 4.1.2.2

Add $0$ and $0$ .

$0$

Step 4.2

Evaluate at $x = - 4$ .

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Step 4.2.1

Substitute $- 4$ for $x$ .

$\frac{1}{4} \cdot {(- 4)}^{4} - 8 {(- 4)}^{2}$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Simplify each term.

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Step 4.2.2.1.1

Raise $- 4$ to the power of $4$ .

$\frac{1}{4} \cdot 256 - 8 {(- 4)}^{2}$

Step 4.2.2.1.2

Cancel the common factor of $4$ .

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Step 4.2.2.1.2.1

Factor $4$ out of $256$ .

$\frac{1}{4} \cdot (4 (64)) - 8 {(- 4)}^{2}$

Step 4.2.2.1.2.2

Cancel the common factor.

$\frac{1}{4} \cdot (4 \cdot 64) - 8 {(- 4)}^{2}$

Step 4.2.2.1.2.3

Rewrite the expression.

$64 - 8 {(- 4)}^{2}$

Step 4.2.2.1.3

Raise $- 4$ to the power of $2$ .

$64 - 8 \cdot 16$

Step 4.2.2.1.4

Multiply $- 8$ by $16$ .

$64 - 128$

Step 4.2.2.2

Subtract $128$ from $64$ .

$- 64$

Step 4.3

Evaluate at $x = 4$ .

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Step 4.3.1

Substitute $4$ for $x$ .

$\frac{1}{4} \cdot {(4)}^{4} - 8 {(4)}^{2}$

Step 4.3.2

Simplify.

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Step 4.3.2.1

Simplify each term.

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Step 4.3.2.1.1

Cancel the common factor of $4$ .

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Step 4.3.2.1.1.1

Factor $4$ out of ${(4)}^{4}$ .

$\frac{1}{4} \cdot (4 \cdot 4^{3}) - 8 {(4)}^{2}$

Step 4.3.2.1.1.2

Cancel the common factor.

$\frac{1}{4} \cdot (4 \cdot 4^{3}) - 8 {(4)}^{2}$

Step 4.3.2.1.1.3

Rewrite the expression.

$4^{3} - 8 {(4)}^{2}$

Step 4.3.2.1.2

Raise $4$ to the power of $3$ .

$64 - 8 {(4)}^{2}$

Step 4.3.2.1.3

Raise $4$ to the power of $2$ .

$64 - 8 \cdot 16$

Step 4.3.2.1.4

Multiply $- 8$ by $16$ .

$64 - 128$

Step 4.3.2.2

Subtract $128$ from $64$ .

$- 64$

Step 4.4

List all of the points.

$(0, 0), (- 4, - 64), (4, - 64)$

Step 5