Calculus Examples

$f (x) = - (\frac{3}{5}) x^{5} - 2 x^{3} + 3 x - 12$ , $[- 4, 3]$

Step 1

Find the critical points.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Find the first derivative.

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Step 1.1.1.1

By the Sum Rule, the derivative of $- \frac{3}{5} x^{5} - 2 x^{3} + 3 x - 12$ with respect to $x$ is $\frac{d}{d x} [- \frac{3}{5} x^{5}] + \frac{d}{d x} [- 2 x^{3}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 12]$ .

$\frac{d}{d x} [- \frac{3}{5} x^{5}] + \frac{d}{d x} [- 2 x^{3}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 12]$

Step 1.1.1.2

Evaluate $\frac{d}{d x} [- \frac{3}{5} x^{5}]$ .

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Step 1.1.1.2.1

Since $- \frac{3}{5}$ is constant with respect to $x$ , the derivative of $- \frac{3}{5} x^{5}$ with respect to $x$ is $- \frac{3}{5} \frac{d}{d x} [x^{5}]$ .

$- \frac{3}{5} \frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 2 x^{3}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 12]$

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 $n = 5$ .

$- \frac{3}{5} (5 x^{4}) + \frac{d}{d x} [- 2 x^{3}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 12]$

Step 1.1.1.2.3

Multiply $5$ by $- 1$ .

$- 5 (\frac{3}{5}) x^{4} + \frac{d}{d x} [- 2 x^{3}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 12]$

Step 1.1.1.2.4

Combine $- 5$ and $\frac{3}{5}$ .

$\frac{- 5 \cdot 3}{5} x^{4} + \frac{d}{d x} [- 2 x^{3}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 12]$

Step 1.1.1.2.5

Multiply $- 5$ by $3$ .

$\frac{- 15}{5} x^{4} + \frac{d}{d x} [- 2 x^{3}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 12]$

Step 1.1.1.2.6

Combine $\frac{- 15}{5}$ and $x^{4}$ .

$\frac{- 15 x^{4}}{5} + \frac{d}{d x} [- 2 x^{3}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 12]$

Step 1.1.1.2.7

Cancel the common factor of $- 15$ and $5$ .

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Step 1.1.1.2.7.1

Factor $5$ out of $- 15 x^{4}$ .

$\frac{5 (- 3 x^{4})}{5} + \frac{d}{d x} [- 2 x^{3}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 12]$

Step 1.1.1.2.7.2

Cancel the common factors.

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Step 1.1.1.2.7.2.1

Factor $5$ out of $5$ .

$\frac{5 (- 3 x^{4})}{5 (1)} + \frac{d}{d x} [- 2 x^{3}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 12]$

Step 1.1.1.2.7.2.2

Cancel the common factor.

$\frac{5 (- 3 x^{4})}{5 \cdot 1} + \frac{d}{d x} [- 2 x^{3}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 12]$

Step 1.1.1.2.7.2.3

Rewrite the expression.

$\frac{- 3 x^{4}}{1} + \frac{d}{d x} [- 2 x^{3}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 12]$

Step 1.1.1.2.7.2.4

Divide $- 3 x^{4}$ by $1$ .

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

Step 1.1.1.3

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

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Step 1.1.1.3.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x^{3}$ with respect to $x$ is $- 2 \frac{d}{d x} [x^{3}]$ .

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

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 $n = 3$ .

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

Step 1.1.1.3.3

Multiply $3$ by $- 2$ .

$- 3 x^{4} - 6 x^{2} + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 12]$

Step 1.1.1.4

Evaluate $\frac{d}{d x} [3 x]$ .

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Step 1.1.1.4.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$- 3 x^{4} - 6 x^{2} + 3 \frac{d}{d x} [x] + \frac{d}{d x} [- 12]$

Step 1.1.1.4.2

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

$- 3 x^{4} - 6 x^{2} + 3 \cdot 1 + \frac{d}{d x} [- 12]$

Step 1.1.1.4.3

Multiply $3$ by $1$ .

$- 3 x^{4} - 6 x^{2} + 3 + \frac{d}{d x} [- 12]$

Step 1.1.1.5

Differentiate using the Constant Rule.

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Step 1.1.1.5.1

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

$- 3 x^{4} - 6 x^{2} + 3 + 0$

Step 1.1.1.5.2

Add $- 3 x^{4} - 6 x^{2} + 3$ and $0$ .

$f' (x) = - 3 x^{4} - 6 x^{2} + 3$

Step 1.1.2

The first derivative of $f (x)$ with respect to $x$ is $- 3 x^{4} - 6 x^{2} + 3$ .

$- 3 x^{4} - 6 x^{2} + 3$

Step 1.2

Set the first derivative equal to $0$ then solve the equation $- 3 x^{4} - 6 x^{2} + 3 = 0$ .

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Step 1.2.1

Set the first derivative equal to $0$ .

$- 3 x^{4} - 6 x^{2} + 3 = 0$

Step 1.2.2

Substitute $u = x^{2}$ into the equation. This will make the quadratic formula easy to use.

$- 3 u^{2} - 6 u + 3 = 0$

$u = x^{2}$

Step 1.2.3

Factor $- 3$ out of $- 3 u^{2} - 6 u + 3$ .

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Step 1.2.3.1

Factor $- 3$ out of $- 3 u^{2}$ .

$- 3 u^{2} - 6 u + 3 = 0$

Step 1.2.3.2

Factor $- 3$ out of $- 6 u$ .

$- 3 u^{2} - 3 (2 u) + 3 = 0$

Step 1.2.3.3

Factor $- 3$ out of $3$ .

$- 3 u^{2} - 3 (2 u) - 3 \cdot - 1 = 0$

Step 1.2.3.4

Factor $- 3$ out of $- 3 (u^{2}) - 3 (2 u)$ .

$- 3 (u^{2} + 2 u) - 3 \cdot - 1 = 0$

Step 1.2.3.5

Factor $- 3$ out of $- 3 (u^{2} + 2 u) - 3 (- 1)$ .

$- 3 (u^{2} + 2 u - 1) = 0$

Step 1.2.4

Divide each term in $- 3 (u^{2} + 2 u - 1) = 0$ by $- 3$ and simplify.

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Step 1.2.4.1

Divide each term in $- 3 (u^{2} + 2 u - 1) = 0$ by $- 3$ .

$\frac{- 3 (u^{2} + 2 u - 1)}{- 3} = \frac{0}{- 3}$

Step 1.2.4.2

Simplify the left side.

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Step 1.2.4.2.1

Cancel the common factor of $- 3$ .

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Step 1.2.4.2.1.1

Cancel the common factor.

$\frac{- 3 (u^{2} + 2 u - 1)}{- 3} = \frac{0}{- 3}$

Step 1.2.4.2.1.2

Divide $u^{2} + 2 u - 1$ by $1$ .

$u^{2} + 2 u - 1 = \frac{0}{- 3}$

Step 1.2.4.3

Simplify the right side.

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Step 1.2.4.3.1

Divide $0$ by $- 3$ .

$u^{2} + 2 u - 1 = 0$

Step 1.2.5

Use the quadratic formula to find the solutions.

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

Step 1.2.6

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

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

Step 1.2.7

Simplify.

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Step 1.2.7.1

Simplify the numerator.

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Step 1.2.7.1.1

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

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

Step 1.2.7.1.2

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

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Step 1.2.7.1.2.1

Multiply $- 4$ by $1$ .

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

Step 1.2.7.1.2.2

Multiply $- 4$ by $- 1$ .

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

Step 1.2.7.1.3

Add $4$ and $4$ .

$u = \frac{- 2 \pm \sqrt{8}}{2 \cdot 1}$

Step 1.2.7.1.4

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 1.2.7.1.4.1

Factor $4$ out of $8$ .

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

Step 1.2.7.1.4.2

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

$u = \frac{- 2 \pm \sqrt{2^{2} \cdot 2}}{2 \cdot 1}$

Step 1.2.7.1.5

Pull terms out from under the radical.

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

Step 1.2.7.2

Multiply $2$ by $1$ .

$u = \frac{- 2 \pm 2 \sqrt{2}}{2}$

Step 1.2.7.3

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

$u = - 1 \pm \sqrt{2}$

Step 1.2.8

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

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Step 1.2.8.1

Simplify the numerator.

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Step 1.2.8.1.1

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

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

Step 1.2.8.1.2

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

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Step 1.2.8.1.2.1

Multiply $- 4$ by $1$ .

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

Step 1.2.8.1.2.2

Multiply $- 4$ by $- 1$ .

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

Step 1.2.8.1.3

Add $4$ and $4$ .

$u = \frac{- 2 \pm \sqrt{8}}{2 \cdot 1}$

Step 1.2.8.1.4

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 1.2.8.1.4.1

Factor $4$ out of $8$ .

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

Step 1.2.8.1.4.2

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

$u = \frac{- 2 \pm \sqrt{2^{2} \cdot 2}}{2 \cdot 1}$

Step 1.2.8.1.5

Pull terms out from under the radical.

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

Step 1.2.8.2

Multiply $2$ by $1$ .

$u = \frac{- 2 \pm 2 \sqrt{2}}{2}$

Step 1.2.8.3

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

$u = - 1 \pm \sqrt{2}$

Step 1.2.8.4

Change the $\pm$ to $+$ .

$u = - 1 + \sqrt{2}$

Step 1.2.9

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

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Step 1.2.9.1

Simplify the numerator.

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Step 1.2.9.1.1

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

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

Step 1.2.9.1.2

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

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Step 1.2.9.1.2.1

Multiply $- 4$ by $1$ .

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

Step 1.2.9.1.2.2

Multiply $- 4$ by $- 1$ .

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

Step 1.2.9.1.3

Add $4$ and $4$ .

$u = \frac{- 2 \pm \sqrt{8}}{2 \cdot 1}$

Step 1.2.9.1.4

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 1.2.9.1.4.1

Factor $4$ out of $8$ .

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

Step 1.2.9.1.4.2

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

$u = \frac{- 2 \pm \sqrt{2^{2} \cdot 2}}{2 \cdot 1}$

Step 1.2.9.1.5

Pull terms out from under the radical.

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

Step 1.2.9.2

Multiply $2$ by $1$ .

$u = \frac{- 2 \pm 2 \sqrt{2}}{2}$

Step 1.2.9.3

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

$u = - 1 \pm \sqrt{2}$

Step 1.2.9.4

Change the $\pm$ to $-$ .

$u = - 1 - \sqrt{2}$

Step 1.2.10

The final answer is the combination of both solutions.

$u = - 1 + \sqrt{2}, - 1 - \sqrt{2}$

Step 1.2.11

Substitute the real value of $u = x^{2}$ back into the solved equation.

$x^{2} = 0.41421356$

${(x^{2})}^{1} = - 2.41421356$

Step 1.2.12

Solve the first equation for $x$ .

$x^{2} = 0.41421356$

Step 1.2.13

Solve the equation for $x$ .

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Step 1.2.13.1

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

$x = \pm \sqrt{0.41421356}$

Step 1.2.13.2

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

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Step 1.2.13.2.1

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

$x = \sqrt{0.41421356}$

Step 1.2.13.2.2

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

$x = - \sqrt{0.41421356}$

Step 1.2.13.2.3

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

$x = \sqrt{0.41421356}, - \sqrt{0.41421356}$

Step 1.2.14

Solve the second equation for $x$ .

${(x^{2})}^{1} = - 2.41421356$

Step 1.2.15

Solve the equation for $x$ .

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Step 1.2.15.1

Remove parentheses.

$x^{2} = - 2.41421356$

Step 1.2.15.2

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

$x = \pm \sqrt{- 2.41421356}$

Step 1.2.15.3

Simplify $\pm \sqrt{- 2.41421356}$ .

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Step 1.2.15.3.1

Rewrite $- 2.41421356$ as $- 1 (2.41421356)$ .

$x = \pm \sqrt{- 1 (2.41421356)}$

Step 1.2.15.3.2

Rewrite $\sqrt{- 1 (2.41421356)}$ as $\sqrt{- 1} \cdot \sqrt{2.41421356}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{2.41421356}$

Step 1.2.15.3.3

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

$x = \pm i \sqrt{2.41421356}$

Step 1.2.15.4

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

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Step 1.2.15.4.1

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

$x = i \sqrt{2.41421356}$

Step 1.2.15.4.2

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

$x = - i \sqrt{2.41421356}$

Step 1.2.15.4.3

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

$x = i \sqrt{2.41421356}, - i \sqrt{2.41421356}$

Step 1.2.16

The solution to $- 3 x^{4} - 6 x^{2} + 3 = 0$ is $x = \sqrt{0.41421356}, - \sqrt{0.41421356}, i \sqrt{2.41421356}, - i \sqrt{2.41421356}$ .

$x = \sqrt{0.41421356}, - \sqrt{0.41421356}, i \sqrt{2.41421356}, - i \sqrt{2.41421356}$

Step 1.3

Find the values where the derivative is undefined.

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Step 1.3.1

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 1.4

Evaluate $- \frac{3}{5} x^{5} - 2 x^{3} + 3 x - 12$ at each $x$ value where the derivative is $0$ or undefined.

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Step 1.4.1

Evaluate at $x = \sqrt{0.41421356}$ .

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Step 1.4.1.1

Substitute $\sqrt{0.41421356}$ for $x$ .

$- \frac{3}{5} \cdot {(\sqrt{0.41421356})}^{5} - 2 {(\sqrt{0.41421356})}^{3} + 3 (\sqrt{0.41421356}) - 12$

Step 1.4.1.2

Simplify each term.

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Step 1.4.1.2.1

Rewrite ${\sqrt{0.41421356}}^{5}$ as $\sqrt{{0.41421356}^{5}}$ .

$- \frac{3}{5} \cdot \sqrt{{0.41421356}^{5}} - 2 {(\sqrt{0.41421356})}^{3} + 3 (\sqrt{0.41421356}) - 12$

Step 1.4.1.2.2

Raise $0.41421356$ to the power of $5$ .

$- \frac{3}{5} \cdot \sqrt{0.0121933} - 2 {(\sqrt{0.41421356})}^{3} + 3 (\sqrt{0.41421356}) - 12$

Step 1.4.1.2.3

Combine $\sqrt{0.0121933}$ and $\frac{3}{5}$ .

$- \frac{\sqrt{0.0121933} \cdot 3}{5} - 2 {(\sqrt{0.41421356})}^{3} + 3 (\sqrt{0.41421356}) - 12$

Step 1.4.1.2.4

Move $3$ to the left of $\sqrt{0.0121933}$ .

$- \frac{3 \sqrt{0.0121933}}{5} - 2 {(\sqrt{0.41421356})}^{3} + 3 (\sqrt{0.41421356}) - 12$

Step 1.4.1.2.5

Rewrite ${\sqrt{0.41421356}}^{3}$ as $\sqrt{{0.41421356}^{3}}$ .

$- \frac{3 \sqrt{0.0121933}}{5} - 2 \sqrt{{0.41421356}^{3}} + 3 (\sqrt{0.41421356}) - 12$

Step 1.4.1.2.6

Raise $0.41421356$ to the power of $3$ .

$- \frac{3 \sqrt{0.0121933}}{5} - 2 \sqrt{0.07106781} + 3 \sqrt{0.41421356} - 12$

Step 1.4.2

Evaluate at $x = - \sqrt{0.41421356}$ .

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Step 1.4.2.1

Substitute $- \sqrt{0.41421356}$ for $x$ .

$- \frac{3}{5} \cdot {(- \sqrt{0.41421356})}^{5} - 2 {(- \sqrt{0.41421356})}^{3} + 3 (- \sqrt{0.41421356}) - 12$

Step 1.4.2.2

Simplify each term.

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Step 1.4.2.2.1

Apply the product rule to $- \sqrt{0.41421356}$ .

$- \frac{3}{5} \cdot ({(- 1)}^{5} {\sqrt{0.41421356}}^{5}) - 2 {(- \sqrt{0.41421356})}^{3} + 3 (- \sqrt{0.41421356}) - 12$

Step 1.4.2.2.2

Multiply $- 1$ by ${(- 1)}^{5}$ by adding the exponents.

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Step 1.4.2.2.2.1

Move ${(- 1)}^{5}$ .

${(- 1)}^{5} \cdot - 1 \frac{3}{5} \cdot {\sqrt{0.41421356}}^{5} - 2 {(- \sqrt{0.41421356})}^{3} + 3 (- \sqrt{0.41421356}) - 12$

Step 1.4.2.2.2.2

Multiply ${(- 1)}^{5}$ by $- 1$ .

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Step 1.4.2.2.2.2.1

Raise $- 1$ to the power of $1$ .

${(- 1)}^{5} \cdot {(- 1)}^{1} \frac{3}{5} \cdot {\sqrt{0.41421356}}^{5} - 2 {(- \sqrt{0.41421356})}^{3} + 3 (- \sqrt{0.41421356}) - 12$

Step 1.4.2.2.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

${(- 1)}^{5 + 1} \frac{3}{5} \cdot {\sqrt{0.41421356}}^{5} - 2 {(- \sqrt{0.41421356})}^{3} + 3 (- \sqrt{0.41421356}) - 12$

Step 1.4.2.2.2.3

Add $5$ and $1$ .

${(- 1)}^{6} \frac{3}{5} \cdot {\sqrt{0.41421356}}^{5} - 2 {(- \sqrt{0.41421356})}^{3} + 3 (- \sqrt{0.41421356}) - 12$

Step 1.4.2.2.3

Raise $- 1$ to the power of $6$ .

$1 (\frac{3}{5}) \cdot {\sqrt{0.41421356}}^{5} - 2 {(- \sqrt{0.41421356})}^{3} + 3 (- \sqrt{0.41421356}) - 12$

Step 1.4.2.2.4

Multiply $\frac{3}{5}$ by $1$ .

$\frac{3}{5} \cdot {\sqrt{0.41421356}}^{5} - 2 {(- \sqrt{0.41421356})}^{3} + 3 (- \sqrt{0.41421356}) - 12$

Step 1.4.2.2.5

Rewrite ${\sqrt{0.41421356}}^{5}$ as $\sqrt{{0.41421356}^{5}}$ .

$\frac{3}{5} \cdot \sqrt{{0.41421356}^{5}} - 2 {(- \sqrt{0.41421356})}^{3} + 3 (- \sqrt{0.41421356}) - 12$

Step 1.4.2.2.6

Raise $0.41421356$ to the power of $5$ .

$\frac{3}{5} \cdot \sqrt{0.0121933} - 2 {(- \sqrt{0.41421356})}^{3} + 3 (- \sqrt{0.41421356}) - 12$

Step 1.4.2.2.7

Combine $\frac{3}{5}$ and $\sqrt{0.0121933}$ .

$\frac{3 \sqrt{0.0121933}}{5} - 2 {(- \sqrt{0.41421356})}^{3} + 3 (- \sqrt{0.41421356}) - 12$

Step 1.4.2.2.8

Apply the product rule to $- \sqrt{0.41421356}$ .

$\frac{3 \sqrt{0.0121933}}{5} - 2 ({(- 1)}^{3} {\sqrt{0.41421356}}^{3}) + 3 (- \sqrt{0.41421356}) - 12$

Step 1.4.2.2.9

Raise $- 1$ to the power of $3$ .

$\frac{3 \sqrt{0.0121933}}{5} - 2 (- {\sqrt{0.41421356}}^{3}) + 3 (- \sqrt{0.41421356}) - 12$

Step 1.4.2.2.10

Rewrite ${\sqrt{0.41421356}}^{3}$ as $\sqrt{{0.41421356}^{3}}$ .

$\frac{3 \sqrt{0.0121933}}{5} - 2 (- \sqrt{{0.41421356}^{3}}) + 3 (- \sqrt{0.41421356}) - 12$

Step 1.4.2.2.11

Raise $0.41421356$ to the power of $3$ .

$\frac{3 \sqrt{0.0121933}}{5} - 2 (- \sqrt{0.07106781}) + 3 (- \sqrt{0.41421356}) - 12$

Step 1.4.2.2.12

Multiply $- 1$ by $- 2$ .

$\frac{3 \sqrt{0.0121933}}{5} + 2 \sqrt{0.07106781} + 3 (- \sqrt{0.41421356}) - 12$

Step 1.4.2.2.13

Multiply $- 1$ by $3$ .

$\frac{3 \sqrt{0.0121933}}{5} + 2 \sqrt{0.07106781} - 3 \sqrt{0.41421356} - 12$

Step 1.4.3

List all of the points.

$(\sqrt{0.41421356}, - \frac{3 \sqrt{0.0121933}}{5} - 2 \sqrt{0.07106781} + 3 \sqrt{0.41421356} - 12), (- \sqrt{0.41421356}, \frac{3 \sqrt{0.0121933}}{5} + 2 \sqrt{0.07106781} - 3 \sqrt{0.41421356} - 12)$

Step 2

Evaluate at the included endpoints.

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Step 2.1

Evaluate at $x = - 4$ .

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Step 2.1.1

Substitute $- 4$ for $x$ .

$- \frac{3}{5} \cdot {(- 4)}^{5} - 2 {(- 4)}^{3} + 3 (- 4) - 12$

Step 2.1.2

Simplify.

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Step 2.1.2.1

Simplify each term.

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Step 2.1.2.1.1

Raise $- 4$ to the power of $5$ .

$- \frac{3}{5} \cdot - 1024 - 2 {(- 4)}^{3} + 3 (- 4) - 12$

Step 2.1.2.1.2

Multiply $- \frac{3}{5} \cdot - 1024$ .

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Step 2.1.2.1.2.1

Multiply $- 1024$ by $- 1$ .

$1024 (\frac{3}{5}) - 2 {(- 4)}^{3} + 3 (- 4) - 12$

Step 2.1.2.1.2.2

Combine $1024$ and $\frac{3}{5}$ .

$\frac{1024 \cdot 3}{5} - 2 {(- 4)}^{3} + 3 (- 4) - 12$

Step 2.1.2.1.2.3

Multiply $1024$ by $3$ .

$\frac{3072}{5} - 2 {(- 4)}^{3} + 3 (- 4) - 12$

Step 2.1.2.1.3

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

$\frac{3072}{5} - 2 \cdot - 64 + 3 (- 4) - 12$

Step 2.1.2.1.4

Multiply $- 2$ by $- 64$ .

$\frac{3072}{5} + 128 + 3 (- 4) - 12$

Step 2.1.2.1.5

Multiply $3$ by $- 4$ .

$\frac{3072}{5} + 128 - 12 - 12$

Step 2.1.2.2

Find the common denominator.

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Step 2.1.2.2.1

Write $128$ as a fraction with denominator $1$ .

$\frac{3072}{5} + \frac{128}{1} - 12 - 12$

Step 2.1.2.2.2

Multiply $\frac{128}{1}$ by $\frac{5}{5}$ .

$\frac{3072}{5} + \frac{128}{1} \cdot \frac{5}{5} - 12 - 12$

Step 2.1.2.2.3

Multiply $\frac{128}{1}$ by $\frac{5}{5}$ .

$\frac{3072}{5} + \frac{128 \cdot 5}{5} - 12 - 12$

Step 2.1.2.2.4

Write $- 12$ as a fraction with denominator $1$ .

$\frac{3072}{5} + \frac{128 \cdot 5}{5} + \frac{- 12}{1} - 12$

Step 2.1.2.2.5

Multiply $\frac{- 12}{1}$ by $\frac{5}{5}$ .

$\frac{3072}{5} + \frac{128 \cdot 5}{5} + \frac{- 12}{1} \cdot \frac{5}{5} - 12$

Step 2.1.2.2.6

Multiply $\frac{- 12}{1}$ by $\frac{5}{5}$ .

$\frac{3072}{5} + \frac{128 \cdot 5}{5} + \frac{- 12 \cdot 5}{5} - 12$

Step 2.1.2.2.7

Write $- 12$ as a fraction with denominator $1$ .

$\frac{3072}{5} + \frac{128 \cdot 5}{5} + \frac{- 12 \cdot 5}{5} + \frac{- 12}{1}$

Step 2.1.2.2.8

Multiply $\frac{- 12}{1}$ by $\frac{5}{5}$ .

$\frac{3072}{5} + \frac{128 \cdot 5}{5} + \frac{- 12 \cdot 5}{5} + \frac{- 12}{1} \cdot \frac{5}{5}$

Step 2.1.2.2.9

Multiply $\frac{- 12}{1}$ by $\frac{5}{5}$ .

$\frac{3072}{5} + \frac{128 \cdot 5}{5} + \frac{- 12 \cdot 5}{5} + \frac{- 12 \cdot 5}{5}$

Step 2.1.2.3

Combine the numerators over the common denominator.

$\frac{3072 + 128 \cdot 5 - 12 \cdot 5 - 12 \cdot 5}{5}$

Step 2.1.2.4

Simplify each term.

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Step 2.1.2.4.1

Multiply $128$ by $5$ .

$\frac{3072 + 640 - 12 \cdot 5 - 12 \cdot 5}{5}$

Step 2.1.2.4.2

Multiply $- 12$ by $5$ .

$\frac{3072 + 640 - 60 - 12 \cdot 5}{5}$

Step 2.1.2.4.3

Multiply $- 12$ by $5$ .

$\frac{3072 + 640 - 60 - 60}{5}$

Step 2.1.2.5

Simplify by adding and subtracting.

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Step 2.1.2.5.1

Add $3072$ and $640$ .

$\frac{3712 - 60 - 60}{5}$

Step 2.1.2.5.2

Subtract $60$ from $3712$ .

$\frac{3652 - 60}{5}$

Step 2.1.2.5.3

Subtract $60$ from $3652$ .

$\frac{3592}{5}$

Step 2.2

Evaluate at $x = 3$ .

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Step 2.2.1

Substitute $3$ for $x$ .

$- \frac{3}{5} \cdot {(3)}^{5} - 2 {(3)}^{3} + 3 (3) - 12$

Step 2.2.2

Simplify.

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Step 2.2.2.1

Simplify each term.

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Step 2.2.2.1.1

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

$- \frac{3}{5} \cdot 243 - 2 {(3)}^{3} + 3 (3) - 12$

Step 2.2.2.1.2

Multiply $- \frac{3}{5} \cdot 243$ .

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Step 2.2.2.1.2.1

Multiply $243$ by $- 1$ .

$- 243 (\frac{3}{5}) - 2 {(3)}^{3} + 3 (3) - 12$

Step 2.2.2.1.2.2

Combine $- 243$ and $\frac{3}{5}$ .

$\frac{- 243 \cdot 3}{5} - 2 {(3)}^{3} + 3 (3) - 12$

Step 2.2.2.1.2.3

Multiply $- 243$ by $3$ .

$\frac{- 729}{5} - 2 {(3)}^{3} + 3 (3) - 12$

Step 2.2.2.1.3

Move the negative in front of the fraction.

$- \frac{729}{5} - 2 {(3)}^{3} + 3 (3) - 12$

Step 2.2.2.1.4

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

$- \frac{729}{5} - 2 \cdot 27 + 3 (3) - 12$

Step 2.2.2.1.5

Multiply $- 2$ by $27$ .

$- \frac{729}{5} - 54 + 3 (3) - 12$

Step 2.2.2.1.6

Multiply $3$ by $3$ .

$- \frac{729}{5} - 54 + 9 - 12$

Step 2.2.2.2

Find the common denominator.

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Step 2.2.2.2.1

Write $- 54$ as a fraction with denominator $1$ .

$- \frac{729}{5} + \frac{- 54}{1} + 9 - 12$

Step 2.2.2.2.2

Multiply $\frac{- 54}{1}$ by $\frac{5}{5}$ .

$- \frac{729}{5} + \frac{- 54}{1} \cdot \frac{5}{5} + 9 - 12$

Step 2.2.2.2.3

Multiply $\frac{- 54}{1}$ by $\frac{5}{5}$ .

$- \frac{729}{5} + \frac{- 54 \cdot 5}{5} + 9 - 12$

Step 2.2.2.2.4

Write $9$ as a fraction with denominator $1$ .

$- \frac{729}{5} + \frac{- 54 \cdot 5}{5} + \frac{9}{1} - 12$

Step 2.2.2.2.5

Multiply $\frac{9}{1}$ by $\frac{5}{5}$ .

$- \frac{729}{5} + \frac{- 54 \cdot 5}{5} + \frac{9}{1} \cdot \frac{5}{5} - 12$

Step 2.2.2.2.6

Multiply $\frac{9}{1}$ by $\frac{5}{5}$ .

$- \frac{729}{5} + \frac{- 54 \cdot 5}{5} + \frac{9 \cdot 5}{5} - 12$

Step 2.2.2.2.7

Write $- 12$ as a fraction with denominator $1$ .

$- \frac{729}{5} + \frac{- 54 \cdot 5}{5} + \frac{9 \cdot 5}{5} + \frac{- 12}{1}$

Step 2.2.2.2.8

Multiply $\frac{- 12}{1}$ by $\frac{5}{5}$ .

$- \frac{729}{5} + \frac{- 54 \cdot 5}{5} + \frac{9 \cdot 5}{5} + \frac{- 12}{1} \cdot \frac{5}{5}$

Step 2.2.2.2.9

Multiply $\frac{- 12}{1}$ by $\frac{5}{5}$ .

$- \frac{729}{5} + \frac{- 54 \cdot 5}{5} + \frac{9 \cdot 5}{5} + \frac{- 12 \cdot 5}{5}$

Step 2.2.2.3

Combine the numerators over the common denominator.

$\frac{- 729 - 54 \cdot 5 + 9 \cdot 5 - 12 \cdot 5}{5}$

Step 2.2.2.4

Simplify each term.

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Step 2.2.2.4.1

Multiply $- 54$ by $5$ .

$\frac{- 729 - 270 + 9 \cdot 5 - 12 \cdot 5}{5}$

Step 2.2.2.4.2

Multiply $9$ by $5$ .

$\frac{- 729 - 270 + 45 - 12 \cdot 5}{5}$

Step 2.2.2.4.3

Multiply $- 12$ by $5$ .

$\frac{- 729 - 270 + 45 - 60}{5}$

Step 2.2.2.5

Simplify the expression.

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Step 2.2.2.5.1

Subtract $270$ from $- 729$ .

$\frac{- 999 + 45 - 60}{5}$

Step 2.2.2.5.2

Add $- 999$ and $45$ .

$\frac{- 954 - 60}{5}$

Step 2.2.2.5.3

Subtract $60$ from $- 954$ .

$\frac{- 1014}{5}$

Step 2.2.2.5.4

Move the negative in front of the fraction.

$- \frac{1014}{5}$

Step 2.3

List all of the points.

$(- 4, \frac{3592}{5}), (3, - \frac{1014}{5})$

Step 3

Compare the $f (x)$ values found for each value of $x$ in order to determine the absolute maximum and minimum over the given interval. The maximum will occur at the highest $f (x)$ value and the minimum will occur at the lowest $f (x)$ value.

Absolute Maximum: $(- 4, \frac{3592}{5})$

Absolute Minimum: $(3, - \frac{1014}{5})$

Step 4