Calculus Examples

$f (x) = - x + \cos (3 π x)$ , $[0, \frac{π}{6}]$

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 $- x + \cos (3 π x)$ with respect to $x$ is $\frac{d}{d x} [- x] + \frac{d}{d x} [\cos (3 π x)]$ .

$\frac{d}{d x} [- x] + \frac{d}{d x} [\cos (3 π x)]$

Step 1.1.1.2

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

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Step 1.1.1.2.1

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

$- \frac{d}{d x} [x] + \frac{d}{d x} [\cos (3 π x)]$

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 $n = 1$ .

$- 1 \cdot 1 + \frac{d}{d x} [\cos (3 π x)]$

Step 1.1.1.2.3

Multiply $- 1$ by $1$ .

$- 1 + \frac{d}{d x} [\cos (3 π x)]$

Step 1.1.1.3

Evaluate $\frac{d}{d x} [\cos (3 π x)]$ .

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Step 1.1.1.3.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos (x)$ and $g (x) = 3 π x$ .

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Step 1.1.1.3.1.1

To apply the Chain Rule, set $u$ as $3 π x$ .

$- 1 + \frac{d}{d u} [\cos (u)] \frac{d}{d x} [3 π x]$

Step 1.1.1.3.1.2

The derivative of $\cos (u)$ with respect to $u$ is $- \sin (u)$ .

$- 1 - \sin (u) \frac{d}{d x} [3 π x]$

Step 1.1.1.3.1.3

Replace all occurrences of $u$ with $3 π x$ .

$- 1 - \sin (3 π x) \frac{d}{d x} [3 π x]$

Step 1.1.1.3.2

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

$- 1 - \sin (3 π x) (3 π \frac{d}{d x} [x])$

Step 1.1.1.3.3

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

$- 1 - \sin (3 π x) (3 π \cdot 1)$

Step 1.1.1.3.4

Multiply $3$ by $1$ .

$- 1 - \sin (3 π x) (3 π)$

Step 1.1.1.3.5

Multiply $3$ by $- 1$ .

$- 1 - 3 \sin (3 π x) π$

Step 1.1.1.4

Reorder terms.

$f' (x) = - 3 π \sin (3 π x) - 1$

Step 1.1.2

The first derivative of $f (x)$ with respect to $x$ is $- 3 π \sin (3 π x) - 1$ .

$- 3 π \sin (3 π x) - 1$

Step 1.2

Set the first derivative equal to $0$ then solve the equation $- 3 π \sin (3 π x) - 1 = 0$ .

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Step 1.2.1

Set the first derivative equal to $0$ .

$- 3 π \sin (3 π x) - 1 = 0$

Step 1.2.2

Add $1$ to both sides of the equation.

$- 3 π \sin (3 π x) = 1$

Step 1.2.3

Divide each term in $- 3 π \sin (3 π x) = 1$ by $- 3 π$ and simplify.

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Step 1.2.3.1

Divide each term in $- 3 π \sin (3 π x) = 1$ by $- 3 π$ .

$\frac{- 3 π \sin (3 π x)}{- 3 π} = \frac{1}{- 3 π}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Cancel the common factor of $- 3$ .

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Step 1.2.3.2.1.1

Cancel the common factor.

$\frac{- 3 π \sin (3 π x)}{- 3 π} = \frac{1}{- 3 π}$

Step 1.2.3.2.1.2

Rewrite the expression.

$\frac{π \sin (3 π x)}{π} = \frac{1}{- 3 π}$

Step 1.2.3.2.2

Cancel the common factor of $π$ .

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Step 1.2.3.2.2.1

Cancel the common factor.

$\frac{π \sin (3 π x)}{π} = \frac{1}{- 3 π}$

Step 1.2.3.2.2.2

Divide $\sin (3 π x)$ by $1$ .

$\sin (3 π x) = \frac{1}{- 3 π}$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Move the negative in front of the fraction.

$\sin (3 π x) = - \frac{1}{3 π}$

Step 1.2.4

Take the inverse sine of both sides of the equation to extract $x$ from inside the sine.

$3 π x = \arcsin (- \frac{1}{3 π})$

Step 1.2.5

Simplify the right side.

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Step 1.2.5.1

Evaluate $\arcsin (- \frac{1}{3 π})$ .

$3 π x = - 0.10630339$

Step 1.2.6

Divide each term in $3 π x = - 0.10630339$ by $3 π$ and simplify.

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Step 1.2.6.1

Divide each term in $3 π x = - 0.10630339$ by $3 π$ .

$\frac{3 π x}{3 π} = \frac{- 0.10630339}{3 π}$

Step 1.2.6.2

Simplify the left side.

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Step 1.2.6.2.1

Cancel the common factor of $3$ .

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Step 1.2.6.2.1.1

Cancel the common factor.

$\frac{3 π x}{3 π} = \frac{- 0.10630339}{3 π}$

Step 1.2.6.2.1.2

Rewrite the expression.

$\frac{π x}{π} = \frac{- 0.10630339}{3 π}$

Step 1.2.6.2.2

Cancel the common factor of $π$ .

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Step 1.2.6.2.2.1

Cancel the common factor.

$\frac{π x}{π} = \frac{- 0.10630339}{3 π}$

Step 1.2.6.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 0.10630339}{3 π}$

Step 1.2.6.3

Simplify the right side.

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Step 1.2.6.3.1

Move the negative in front of the fraction.

$x = - \frac{0.10630339}{3 π}$

Step 1.2.6.3.2

Replace $π$ with an approximation.

$x = - \frac{0.10630339}{3 \cdot 3.14159265}$

Step 1.2.6.3.3

Multiply $3$ by $3.14159265$ .

$x = - \frac{0.10630339}{9.42477796}$

Step 1.2.6.3.4

Divide $0.10630339$ by $9.42477796$ .

$x = - 1 \cdot 0.01127914$

Step 1.2.6.3.5

Multiply $- 1$ by $0.01127914$ .

$x = - 0.01127914$

Step 1.2.7

The sine function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from $2 π$ , to find a reference angle. Next, add this reference angle to $π$ to find the solution in the third quadrant.

$3 π x = 2 (3.14159265) + 0.10630339 + 3.14159265$

Step 1.2.8

Simplify the expression to find the second solution.

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Step 1.2.8.1

Subtract $2 π$ from $2 (3.14159265) + 0.10630339 + 3.14159265$ .

$3 π x = 2 (3.14159265) + 0.10630339 + 3.14159265 - 2 π$

Step 1.2.8.2

The resulting angle of $3.24789604$ is positive, less than $2 π$ , and coterminal with $2 (3.14159265) + 0.10630339 + 3.14159265$ .

$3 π x = 3.24789604$

Step 1.2.8.3

Divide each term in $3 π x = 3.24789604$ by $3 π$ and simplify.

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Step 1.2.8.3.1

Divide each term in $3 π x = 3.24789604$ by $3 π$ .

$\frac{3 π x}{3 π} = \frac{3.24789604}{3 π}$

Step 1.2.8.3.2

Simplify the left side.

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Step 1.2.8.3.2.1

Cancel the common factor of $3$ .

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Step 1.2.8.3.2.1.1

Cancel the common factor.

$\frac{3 π x}{3 π} = \frac{3.24789604}{3 π}$

Step 1.2.8.3.2.1.2

Rewrite the expression.

$\frac{π x}{π} = \frac{3.24789604}{3 π}$

Step 1.2.8.3.2.2

Cancel the common factor of $π$ .

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Step 1.2.8.3.2.2.1

Cancel the common factor.

$\frac{π x}{π} = \frac{3.24789604}{3 π}$

Step 1.2.8.3.2.2.2

Divide $x$ by $1$ .

$x = \frac{3.24789604}{3 π}$

Step 1.2.8.3.3

Simplify the right side.

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Step 1.2.8.3.3.1

Replace $π$ with an approximation.

$x = \frac{3.24789604}{3 \cdot 3.14159265}$

Step 1.2.8.3.3.2

Multiply $3$ by $3.14159265$ .

$x = \frac{3.24789604}{9.42477796}$

Step 1.2.8.3.3.3

Divide $3.24789604$ by $9.42477796$ .

$x = 0.34461247$

Step 1.2.9

Find the period of $\sin (3 π x)$ .

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Step 1.2.9.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 1.2.9.2

Replace $b$ with $3 π$ in the formula for period.

$\frac{2 π}{| 3 π |}$

Step 1.2.9.3

$3 π$ is approximately $9.42477796$ which is positive so remove the absolute value

$\frac{2 π}{3 π}$

Step 1.2.9.4

Cancel the common factor of $π$ .

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Step 1.2.9.4.1

Cancel the common factor.

$\frac{2 π}{3 π}$

Step 1.2.9.4.2

Rewrite the expression.

$\frac{2}{3}$

Step 1.2.10

Add $\frac{2}{3}$ to every negative angle to get positive angles.

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Step 1.2.10.1

Add $\frac{2}{3}$ to $- 0.01127914$ to find the positive angle.

$- 0.01127914 + \frac{2}{3}$

Step 1.2.10.2

To write $- 0.01127914$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{2}{3} - 0.01127914 \cdot \frac{3}{3}$

Step 1.2.10.3

Combine fractions.

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Step 1.2.10.3.1

Combine $- 0.01127914$ and $\frac{3}{3}$ .

$\frac{2}{3} + \frac{- 0.01127914 \cdot 3}{3}$

Step 1.2.10.3.2

Combine the numerators over the common denominator.

$\frac{2 - 0.01127914 \cdot 3}{3}$

Step 1.2.10.4

Simplify the numerator.

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Step 1.2.10.4.1

Multiply $- 0.01127914$ by $3$ .

$\frac{2 - 0.03383742}{3}$

Step 1.2.10.4.2

Subtract $0.03383742$ from $2$ .

$\frac{1.96616257}{3}$

Step 1.2.10.5

Divide $1.96616257$ by $3$ .

$0.65538752$

Step 1.2.10.6

List the new angles.

$x = 0.65538752$

Step 1.2.11

The period of the $\sin (3 π x)$ function is $\frac{2}{3}$ so values will repeat every $\frac{2}{3}$ radians in both directions.

$x = 0.34461247 + \frac{2 n}{3}, 0.65538752 + \frac{2 n}{3}$ , for any integer $n$

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 $- x + \cos (3 π x)$ at each $x$ value where the derivative is $0$ or undefined.

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Step 1.4.1

Evaluate at $x = 0.34461247$ .

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Step 1.4.1.1

Substitute $0.34461247$ for $x$ .

$- (0.34461247) + \cos (3 π (0.34461247))$

Step 1.4.1.2

Simplify.

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Step 1.4.1.2.1

Simplify each term.

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Step 1.4.1.2.1.1

Multiply $- 1$ by $0.34461247$ .

$- 0.34461247 + \cos (3 π (0.34461247))$

Step 1.4.1.2.1.2

Multiply $3 π (0.34461247)$ .

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Step 1.4.1.2.1.2.1

Multiply $0.34461247$ by $3$ .

$- 0.34461247 + \cos (1.03383742 π)$

Step 1.4.1.2.1.2.2

Multiply $1.03383742$ by $π$ .

$- 0.34461247 + \cos (3.24789604)$

Step 1.4.1.2.2

Add $- 0.34461247$ and $\cos (3.24789604)$ .

$- 1.33896758$

Step 1.4.2

Evaluate at $x = 1.01127914$ .

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Step 1.4.2.1

Substitute $1.01127914$ for $x$ .

$- (1.01127914) + \cos (3 π (1.01127914))$

Step 1.4.2.2

Simplify.

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Step 1.4.2.2.1

Simplify each term.

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Step 1.4.2.2.1.1

Multiply $- 1$ by $1.01127914$ .

$- 1.01127914 + \cos (3 π (1.01127914))$

Step 1.4.2.2.1.2

Multiply $3 π (1.01127914)$ .

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Step 1.4.2.2.1.2.1

Multiply $1.01127914$ by $3$ .

$- 1.01127914 + \cos (3.03383742 π)$

Step 1.4.2.2.1.2.2

Multiply $3.03383742$ by $π$ .

$- 1.01127914 + \cos (9.53108135)$

Step 1.4.2.2.2

Add $- 1.01127914$ and $\cos (9.53108135)$ .

$- 2.00563425$

Step 1.4.3

Evaluate at $x = 1.6779458$ .

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Step 1.4.3.1

Substitute $1.6779458$ for $x$ .

$- (1.6779458) + \cos (3 π (1.6779458))$

Step 1.4.3.2

Simplify.

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Step 1.4.3.2.1

Simplify each term.

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Step 1.4.3.2.1.1

Multiply $- 1$ by $1.6779458$ .

$- 1.6779458 + \cos (3 π (1.6779458))$

Step 1.4.3.2.1.2

Multiply $3 π (1.6779458)$ .

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Step 1.4.3.2.1.2.1

Multiply $1.6779458$ by $3$ .

$- 1.6779458 + \cos (5.03383742 π)$

Step 1.4.3.2.1.2.2

Multiply $5.03383742$ by $π$ .

$- 1.6779458 + \cos (15.81426666)$

Step 1.4.3.2.2

Add $- 1.6779458$ and $\cos (15.81426666)$ .

$- 2.67230092$

Step 1.4.4

Evaluate at $x = 2.34461247$ .

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Step 1.4.4.1

Substitute $2.34461247$ for $x$ .

$- (2.34461247) + \cos (3 π (2.34461247))$

Step 1.4.4.2

Simplify.

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Step 1.4.4.2.1

Simplify each term.

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Step 1.4.4.2.1.1

Multiply $- 1$ by $2.34461247$ .

$- 2.34461247 + \cos (3 π (2.34461247))$

Step 1.4.4.2.1.2

Multiply $3 π (2.34461247)$ .

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Step 1.4.4.2.1.2.1

Multiply $2.34461247$ by $3$ .

$- 2.34461247 + \cos (7.03383742 π)$

Step 1.4.4.2.1.2.2

Multiply $7.03383742$ by $π$ .

$- 2.34461247 + \cos (22.09745196)$

Step 1.4.4.2.2

Add $- 2.34461247$ and $\cos (22.09745196)$ .

$- 3.33896758$

Step 1.4.5

Evaluate at $x = 3.01127914$ .

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Step 1.4.5.1

Substitute $3.01127914$ for $x$ .

$- (3.01127914) + \cos (3 π (3.01127914))$

Step 1.4.5.2

Simplify.

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Step 1.4.5.2.1

Simplify each term.

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Step 1.4.5.2.1.1

Multiply $- 1$ by $3.01127914$ .

$- 3.01127914 + \cos (3 π (3.01127914))$

Step 1.4.5.2.1.2

Multiply $3 π (3.01127914)$ .

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Step 1.4.5.2.1.2.1

Multiply $3.01127914$ by $3$ .

$- 3.01127914 + \cos (9.03383742 π)$

Step 1.4.5.2.1.2.2

Multiply $9.03383742$ by $π$ .

$- 3.01127914 + \cos (28.38063727)$

Step 1.4.5.2.2

Add $- 3.01127914$ and $\cos (28.38063727)$ .

$- 4.00563425$

Step 1.4.6

Evaluate at $x = 0.65538752$ .

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Step 1.4.6.1

Substitute $0.65538752$ for $x$ .

$- (0.65538752) + \cos (3 π (0.65538752))$

Step 1.4.6.2

Simplify.

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Step 1.4.6.2.1

Simplify each term.

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Step 1.4.6.2.1.1

Multiply $- 1$ by $0.65538752$ .

$- 0.65538752 + \cos (3 π (0.65538752))$

Step 1.4.6.2.1.2

Multiply $3 π (0.65538752)$ .

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Step 1.4.6.2.1.2.1

Multiply $0.65538752$ by $3$ .

$- 0.65538752 + \cos (1.96616257 π)$

Step 1.4.6.2.1.2.2

Multiply $1.96616257$ by $π$ .

$- 0.65538752 + \cos (6.17688191)$

Step 1.4.6.2.2

Add $- 0.65538752$ and $\cos (6.17688191)$ .

$0.33896758$

Step 1.4.7

Evaluate at $x = 1.32205419$ .

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Step 1.4.7.1

Substitute $1.32205419$ for $x$ .

$- (1.32205419) + \cos (3 π (1.32205419))$

Step 1.4.7.2

Simplify.

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Step 1.4.7.2.1

Simplify each term.

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Step 1.4.7.2.1.1

Multiply $- 1$ by $1.32205419$ .

$- 1.32205419 + \cos (3 π (1.32205419))$

Step 1.4.7.2.1.2

Multiply $3 π (1.32205419)$ .

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Step 1.4.7.2.1.2.1

Multiply $1.32205419$ by $3$ .

$- 1.32205419 + \cos (3.96616257 π)$

Step 1.4.7.2.1.2.2

Multiply $3.96616257$ by $π$ .

$- 1.32205419 + \cos (12.46006722)$

Step 1.4.7.2.2

Add $- 1.32205419$ and $\cos (12.46006722)$ .

$- 0.32769907$

Step 1.4.8

Evaluate at $x = 1.98872085$ .

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Step 1.4.8.1

Substitute $1.98872085$ for $x$ .

$- (1.98872085) + \cos (3 π (1.98872085))$

Step 1.4.8.2

Simplify.

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Step 1.4.8.2.1

Simplify each term.

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Step 1.4.8.2.1.1

Multiply $- 1$ by $1.98872085$ .

$- 1.98872085 + \cos (3 π (1.98872085))$

Step 1.4.8.2.1.2

Multiply $3 π (1.98872085)$ .

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Step 1.4.8.2.1.2.1

Multiply $1.98872085$ by $3$ .

$- 1.98872085 + \cos (5.96616257 π)$

Step 1.4.8.2.1.2.2

Multiply $5.96616257$ by $π$ .

$- 1.98872085 + \cos (18.74325252)$

Step 1.4.8.2.2

Add $- 1.98872085$ and $\cos (18.74325252)$ .

$- 0.99436574$

Step 1.4.9

Evaluate at $x = 2.65538752$ .

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Step 1.4.9.1

Substitute $2.65538752$ for $x$ .

$- (2.65538752) + \cos (3 π (2.65538752))$

Step 1.4.9.2

Simplify.

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Step 1.4.9.2.1

Simplify each term.

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Step 1.4.9.2.1.1

Multiply $- 1$ by $2.65538752$ .

$- 2.65538752 + \cos (3 π (2.65538752))$

Step 1.4.9.2.1.2

Multiply $3 π (2.65538752)$ .

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Step 1.4.9.2.1.2.1

Multiply $2.65538752$ by $3$ .

$- 2.65538752 + \cos (7.96616257 π)$

Step 1.4.9.2.1.2.2

Multiply $7.96616257$ by $π$ .

$- 2.65538752 + \cos (25.02643783)$

Step 1.4.9.2.2

Add $- 2.65538752$ and $\cos (25.02643783)$ .

$- 1.66103241$

Step 1.4.10

Evaluate at $x = 3.32205419$ .

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Step 1.4.10.1

Substitute $3.32205419$ for $x$ .

$- (3.32205419) + \cos (3 π (3.32205419))$

Step 1.4.10.2

Simplify.

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Step 1.4.10.2.1

Simplify each term.

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Step 1.4.10.2.1.1

Multiply $- 1$ by $3.32205419$ .

$- 3.32205419 + \cos (3 π (3.32205419))$

Step 1.4.10.2.1.2

Multiply $3 π (3.32205419)$ .

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Step 1.4.10.2.1.2.1

Multiply $3.32205419$ by $3$ .

$- 3.32205419 + \cos (9.96616257 π)$

Step 1.4.10.2.1.2.2

Multiply $9.96616257$ by $π$ .

$- 3.32205419 + \cos (31.30962314)$

Step 1.4.10.2.2

Add $- 3.32205419$ and $\cos (31.30962314)$ .

$- 2.32769907$

Step 1.4.11

List all of the points.

$(0.34461247, - 1.33896758), (1.01127914, - 2.00563425), (1.6779458, - 2.67230092), (2.34461247, - 3.33896758), (3.01127914, - 4.00563425), (0.65538752, 0.33896758), (1.32205419, - 0.32769907), (1.98872085, - 0.99436574), (2.65538752, - 1.66103241), (3.32205419, - 2.32769907)$

Step 2

Exclude the points that are not on the interval.

$(0.34461247, - 1.33896758)$

Step 3

Evaluate at the included endpoints.

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Step 3.1

Evaluate at $x = 0$ .

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Step 3.1.1

Substitute $0$ for $x$ .

$- (0) + \cos (3 π (0))$

Step 3.1.2

Simplify.

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Step 3.1.2.1

Simplify each term.

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Step 3.1.2.1.1

Multiply $- 1$ by $0$ .

$0 + \cos (3 π (0))$

Step 3.1.2.1.2

Multiply $3 π (0)$ .

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Step 3.1.2.1.2.1

Multiply $0$ by $3$ .

$0 + \cos (0 π)$

Step 3.1.2.1.2.2

Multiply $0$ by $π$ .

$0 + \cos (0)$

Step 3.1.2.1.3

The exact value of $\cos (0)$ is $1$ .

$0 + 1$

Step 3.1.2.2

Add $0$ and $1$ .

$1$

Step 3.2

Evaluate at $x = \frac{π}{6}$ .

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Step 3.2.1

Substitute $\frac{π}{6}$ for $x$ .

$- (\frac{π}{6}) + \cos (3 π (\frac{π}{6}))$

Step 3.2.2

Simplify.

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Step 3.2.2.1

Simplify each term.

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Step 3.2.2.1.1

Cancel the common factor of $3$ .

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Step 3.2.2.1.1.1

Factor $3$ out of $3 π$ .

$- \frac{π}{6} + \cos (3 (π) \frac{π}{6})$

Step 3.2.2.1.1.2

Factor $3$ out of $6$ .

$- \frac{π}{6} + \cos (3 (π) \frac{π}{3 (2)})$

Step 3.2.2.1.1.3

Cancel the common factor.

$- \frac{π}{6} + \cos (3 π \frac{π}{3 \cdot 2})$

Step 3.2.2.1.1.4

Rewrite the expression.

$- \frac{π}{6} + \cos (π \frac{π}{2})$

Step 3.2.2.1.2

Combine $π$ and $\frac{π}{2}$ .

$- \frac{π}{6} + \cos (\frac{π π}{2})$

Step 3.2.2.1.3

Raise $π$ to the power of $1$ .

$- \frac{π}{6} + \cos (\frac{π^{1} π}{2})$

Step 3.2.2.1.4

Raise $π$ to the power of $1$ .

$- \frac{π}{6} + \cos (\frac{π^{1} π^{1}}{2})$

Step 3.2.2.1.5

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

$- \frac{π}{6} + \cos (\frac{π^{1 + 1}}{2})$

Step 3.2.2.1.6

Add $1$ and $1$ .

$- \frac{π}{6} + \cos (\frac{π^{2}}{2})$

Step 3.2.2.1.7

Evaluate $\cos (\frac{π^{2}}{2})$ .

$- \frac{π}{6} + 0.22058404$

Step 3.2.2.2

Add $- \frac{π}{6}$ and $0.22058404$ .

$- 0.30301473$

Step 3.3

List all of the points.

$(0, 1), (\frac{π}{6}, - 0.30301473)$

Step 4

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: $(0, 1)$

Absolute Minimum: $(0.34461247, - 1.33896758)$

Step 5