Calculus Examples

$f (x) = 3 \sin (x) \cos (x)$ , $[\frac{π}{4}, π]$

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

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

$3 \frac{d}{d x} [\sin (x) \cos (x)]$

Step 1.1.1.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = \sin (x)$ and $g (x) = \cos (x)$ .

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

Step 1.1.1.3

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

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

Step 1.1.1.4

Raise $\sin (x)$ to the power of $1$ .

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

Step 1.1.1.5

Raise $\sin (x)$ to the power of $1$ .

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

Step 1.1.1.6

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

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

Step 1.1.1.7

Add $1$ and $1$ .

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

Step 1.1.1.8

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

$3 (- \sin^{2} (x) + \cos (x) \cos (x))$

Step 1.1.1.9

Raise $\cos (x)$ to the power of $1$ .

$3 (- \sin^{2} (x) + \cos^{1} (x) \cos (x))$

Step 1.1.1.10

Raise $\cos (x)$ to the power of $1$ .

$3 (- \sin^{2} (x) + \cos^{1} (x) \cos^{1} (x))$

Step 1.1.1.11

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

$3 (- \sin^{2} (x) + {\cos (x)}^{1 + 1})$

Step 1.1.1.12

Add $1$ and $1$ .

$3 (- \sin^{2} (x) + \cos^{2} (x))$

Step 1.1.1.13

Simplify.

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Step 1.1.1.13.1

Apply the distributive property.

$3 (- \sin^{2} (x)) + 3 \cos^{2} (x)$

Step 1.1.1.13.2

Multiply $- 1$ by $3$ .

$f' (x) = - 3 \sin^{2} (x) + 3 \cos^{2} (x)$

Step 1.1.2

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

$- 3 \sin^{2} (x) + 3 \cos^{2} (x)$

Step 1.2

Set the first derivative equal to $0$ then solve the equation $- 3 \sin^{2} (x) + 3 \cos^{2} (x) = 0$ .

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Step 1.2.1

Set the first derivative equal to $0$ .

$- 3 \sin^{2} (x) + 3 \cos^{2} (x) = 0$

Step 1.2.2

Factor $- 3 \sin^{2} (x) + 3 \cos^{2} (x)$ .

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Step 1.2.2.1

Factor $3$ out of $- 3 \sin^{2} (x) + 3 \cos^{2} (x)$ .

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Step 1.2.2.1.1

Factor $3$ out of $- 3 \sin^{2} (x)$ .

$3 (- \sin^{2} (x)) + 3 \cos^{2} (x) = 0$

Step 1.2.2.1.2

Factor $3$ out of $3 \cos^{2} (x)$ .

$3 (- \sin^{2} (x)) + 3 \cos^{2} (x) = 0$

Step 1.2.2.1.3

Factor $3$ out of $3 (- \sin^{2} (x)) + 3 \cos^{2} (x)$ .

$3 (- \sin^{2} (x) + \cos^{2} (x)) = 0$

Step 1.2.2.2

Reorder $- \sin^{2} (x)$ and $\cos^{2} (x)$ .

$3 (\cos^{2} (x) - \sin^{2} (x)) = 0$

Step 1.2.2.3

Factor.

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Step 1.2.2.3.1

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

$3 ((\cos (x) + \sin (x)) (\cos (x) - \sin (x))) = 0$

Step 1.2.2.3.2

Remove unnecessary parentheses.

$3 (\cos (x) + \sin (x)) (\cos (x) - \sin (x)) = 0$

Step 1.2.3

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

$\cos (x) + \sin (x) = 0$

$\cos (x) - \sin (x) = 0$

Step 1.2.4

Set $\cos (x) + \sin (x)$ equal to $0$ and solve for $x$ .

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Step 1.2.4.1

Set $\cos (x) + \sin (x)$ equal to $0$ .

$\cos (x) + \sin (x) = 0$

Step 1.2.4.2

Solve $\cos (x) + \sin (x) = 0$ for $x$ .

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Step 1.2.4.2.1

Divide each term in the equation by $\cos (x)$ .

$\frac{\cos (x)}{\cos (x)} + \frac{\sin (x)}{\cos (x)} = \frac{0}{\cos (x)}$

Step 1.2.4.2.2

Cancel the common factor of $\cos (x)$ .

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Step 1.2.4.2.2.1

Cancel the common factor.

$\frac{\cos (x)}{\cos (x)} + \frac{\sin (x)}{\cos (x)} = \frac{0}{\cos (x)}$

Step 1.2.4.2.2.2

Rewrite the expression.

$1 + \frac{\sin (x)}{\cos (x)} = \frac{0}{\cos (x)}$

Step 1.2.4.2.3

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$1 + \tan (x) = \frac{0}{\cos (x)}$

Step 1.2.4.2.4

Separate fractions.

$1 + \tan (x) = \frac{0}{1} \cdot \frac{1}{\cos (x)}$

Step 1.2.4.2.5

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$1 + \tan (x) = \frac{0}{1} \cdot \sec (x)$

Step 1.2.4.2.6

Divide $0$ by $1$ .

$1 + \tan (x) = 0 \sec (x)$

Step 1.2.4.2.7

Multiply $0$ by $\sec (x)$ .

$1 + \tan (x) = 0$

Step 1.2.4.2.8

Subtract $1$ from both sides of the equation.

$\tan (x) = - 1$

Step 1.2.4.2.9

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

$x = \arctan (- 1)$

Step 1.2.4.2.10

Simplify the right side.

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Step 1.2.4.2.10.1

The exact value of $\arctan (- 1)$ is $- \frac{π}{4}$ .

$x = - \frac{π}{4}$

Step 1.2.4.2.11

The tangent function is negative in the second and fourth quadrants. To find the second solution, subtract the reference angle from $π$ to find the solution in the third quadrant.

$x = - \frac{π}{4} - π$

Step 1.2.4.2.12

Simplify the expression to find the second solution.

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Step 1.2.4.2.12.1

Add $2 π$ to $- \frac{π}{4} - π$ .

$x = - \frac{π}{4} - π + 2 π$

Step 1.2.4.2.12.2

The resulting angle of $\frac{3 π}{4}$ is positive and coterminal with $- \frac{π}{4} - π$ .

$x = \frac{3 π}{4}$

Step 1.2.4.2.13

Find the period of $\tan (x)$ .

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Step 1.2.4.2.13.1

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

$\frac{π}{| b |}$

Step 1.2.4.2.13.2

Replace $b$ with $1$ in the formula for period.

$\frac{π}{| 1 |}$

Step 1.2.4.2.13.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{π}{1}$

Step 1.2.4.2.13.4

Divide $π$ by $1$ .

$π$

Step 1.2.4.2.14

Add $π$ to every negative angle to get positive angles.

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Step 1.2.4.2.14.1

Add $π$ to $- \frac{π}{4}$ to find the positive angle.

$- \frac{π}{4} + π$

Step 1.2.4.2.14.2

To write $π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$π \cdot \frac{4}{4} - \frac{π}{4}$

Step 1.2.4.2.14.3

Combine fractions.

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Step 1.2.4.2.14.3.1

Combine $π$ and $\frac{4}{4}$ .

$\frac{π \cdot 4}{4} - \frac{π}{4}$

Step 1.2.4.2.14.3.2

Combine the numerators over the common denominator.

$\frac{π \cdot 4 - π}{4}$

Step 1.2.4.2.14.4

Simplify the numerator.

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Step 1.2.4.2.14.4.1

Move $4$ to the left of $π$ .

$\frac{4 \cdot π - π}{4}$

Step 1.2.4.2.14.4.2

Subtract $π$ from $4 π$ .

$\frac{3 π}{4}$

Step 1.2.4.2.14.5

List the new angles.

$x = \frac{3 π}{4}$

Step 1.2.4.2.15

The period of the $\tan (x)$ function is $π$ so values will repeat every $π$ radians in both directions.

$x = \frac{3 π}{4} + π n, \frac{3 π}{4} + π n$ , for any integer $n$

Step 1.2.5

Set $\cos (x) - \sin (x)$ equal to $0$ and solve for $x$ .

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Step 1.2.5.1

Set $\cos (x) - \sin (x)$ equal to $0$ .

$\cos (x) - \sin (x) = 0$

Step 1.2.5.2

Solve $\cos (x) - \sin (x) = 0$ for $x$ .

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Step 1.2.5.2.1

Divide each term in the equation by $\cos (x)$ .

$\frac{\cos (x)}{\cos (x)} + \frac{- \sin (x)}{\cos (x)} = \frac{0}{\cos (x)}$

Step 1.2.5.2.2

Cancel the common factor of $\cos (x)$ .

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Step 1.2.5.2.2.1

Cancel the common factor.

$\frac{\cos (x)}{\cos (x)} + \frac{- \sin (x)}{\cos (x)} = \frac{0}{\cos (x)}$

Step 1.2.5.2.2.2

Rewrite the expression.

$1 + \frac{- \sin (x)}{\cos (x)} = \frac{0}{\cos (x)}$

Step 1.2.5.2.3

Separate fractions.

$1 + \frac{- 1}{1} \cdot \frac{\sin (x)}{\cos (x)} = \frac{0}{\cos (x)}$

Step 1.2.5.2.4

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$1 + \frac{- 1}{1} \cdot \tan (x) = \frac{0}{\cos (x)}$

Step 1.2.5.2.5

Divide $- 1$ by $1$ .

$1 - \tan (x) = \frac{0}{\cos (x)}$

Step 1.2.5.2.6

Separate fractions.

$1 - \tan (x) = \frac{0}{1} \cdot \frac{1}{\cos (x)}$

Step 1.2.5.2.7

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$1 - \tan (x) = \frac{0}{1} \cdot \sec (x)$

Step 1.2.5.2.8

Divide $0$ by $1$ .

$1 - \tan (x) = 0 \sec (x)$

Step 1.2.5.2.9

Multiply $0$ by $\sec (x)$ .

$1 - \tan (x) = 0$

Step 1.2.5.2.10

Subtract $1$ from both sides of the equation.

$- \tan (x) = - 1$

Step 1.2.5.2.11

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

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Step 1.2.5.2.11.1

Divide each term in $- \tan (x) = - 1$ by $- 1$ .

$\frac{- \tan (x)}{- 1} = \frac{- 1}{- 1}$

Step 1.2.5.2.11.2

Simplify the left side.

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Step 1.2.5.2.11.2.1

Dividing two negative values results in a positive value.

$\frac{\tan (x)}{1} = \frac{- 1}{- 1}$

Step 1.2.5.2.11.2.2

Divide $\tan (x)$ by $1$ .

$\tan (x) = \frac{- 1}{- 1}$

Step 1.2.5.2.11.3

Simplify the right side.

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Step 1.2.5.2.11.3.1

Divide $- 1$ by $- 1$ .

$\tan (x) = 1$

Step 1.2.5.2.12

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

$x = \arctan (1)$

Step 1.2.5.2.13

Simplify the right side.

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Step 1.2.5.2.13.1

The exact value of $\arctan (1)$ is $\frac{π}{4}$ .

$x = \frac{π}{4}$

Step 1.2.5.2.14

The tangent function is positive in the first and third quadrants. To find the second solution, add the reference angle from $π$ to find the solution in the fourth quadrant.

$x = π + \frac{π}{4}$

Step 1.2.5.2.15

Simplify $π + \frac{π}{4}$ .

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Step 1.2.5.2.15.1

To write $π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = π \cdot \frac{4}{4} + \frac{π}{4}$

Step 1.2.5.2.15.2

Combine fractions.

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Step 1.2.5.2.15.2.1

Combine $π$ and $\frac{4}{4}$ .

$x = \frac{π \cdot 4}{4} + \frac{π}{4}$

Step 1.2.5.2.15.2.2

Combine the numerators over the common denominator.

$x = \frac{π \cdot 4 + π}{4}$

Step 1.2.5.2.15.3

Simplify the numerator.

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Step 1.2.5.2.15.3.1

Move $4$ to the left of $π$ .

$x = \frac{4 \cdot π + π}{4}$

Step 1.2.5.2.15.3.2

Add $4 π$ and $π$ .

$x = \frac{5 π}{4}$

Step 1.2.5.2.16

Find the period of $\tan (x)$ .

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Step 1.2.5.2.16.1

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

$\frac{π}{| b |}$

Step 1.2.5.2.16.2

Replace $b$ with $1$ in the formula for period.

$\frac{π}{| 1 |}$

Step 1.2.5.2.16.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{π}{1}$

Step 1.2.5.2.16.4

Divide $π$ by $1$ .

$π$

Step 1.2.5.2.17

The period of the $\tan (x)$ function is $π$ so values will repeat every $π$ radians in both directions.

$x = \frac{π}{4} + π n, \frac{5 π}{4} + π n$ , for any integer $n$

Step 1.2.6

The final solution is all the values that make $3 (\cos (x) + \sin (x)) (\cos (x) - \sin (x)) = 0$ true.

$x = \frac{3 π}{4} + π n, \frac{π}{4} + π n, \frac{5 π}{4} + π n$ , for any integer $n$

Step 1.2.7

Consolidate the answers.

$x = \frac{π}{4} + \frac{π n}{2}$ , 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 $3 \sin (x) \cos (x)$ at each $x$ value where the derivative is $0$ or undefined.

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Step 1.4.1

Evaluate at $x = \frac{π}{4}$ .

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Step 1.4.1.1

Substitute $\frac{π}{4}$ for $x$ .

$3 \sin (\frac{π}{4}) \cos (\frac{π}{4})$

Step 1.4.1.2

Simplify.

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Step 1.4.1.2.1

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$3 \frac{\sqrt{2}}{2} \cos (\frac{π}{4})$

Step 1.4.1.2.2

Combine $3$ and $\frac{\sqrt{2}}{2}$ .

$\frac{3 \sqrt{2}}{2} \cos (\frac{π}{4})$

Step 1.4.1.2.3

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$\frac{3 \sqrt{2}}{2} \cdot \frac{\sqrt{2}}{2}$

Step 1.4.1.2.4

Multiply $\frac{3 \sqrt{2}}{2} \cdot \frac{\sqrt{2}}{2}$ .

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Step 1.4.1.2.4.1

Multiply $\frac{3 \sqrt{2}}{2}$ by $\frac{\sqrt{2}}{2}$ .

$\frac{3 \sqrt{2} \sqrt{2}}{2 \cdot 2}$

Step 1.4.1.2.4.2

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{3 ({\sqrt{2}}^{1} \sqrt{2})}{2 \cdot 2}$

Step 1.4.1.2.4.3

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{3 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}{2 \cdot 2}$

Step 1.4.1.2.4.4

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

$\frac{3 {\sqrt{2}}^{1 + 1}}{2 \cdot 2}$

Step 1.4.1.2.4.5

Add $1$ and $1$ .

$\frac{3 {\sqrt{2}}^{2}}{2 \cdot 2}$

Step 1.4.1.2.4.6

Multiply $2$ by $2$ .

$\frac{3 {\sqrt{2}}^{2}}{4}$

Step 1.4.1.2.5

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 1.4.1.2.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

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

Step 1.4.1.2.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{3 \cdot 2^{\frac{1}{2} \cdot 2}}{4}$

Step 1.4.1.2.5.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{3 \cdot 2^{\frac{2}{2}}}{4}$

Step 1.4.1.2.5.4

Cancel the common factor of $2$ .

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Step 1.4.1.2.5.4.1

Cancel the common factor.

$\frac{3 \cdot 2^{\frac{2}{2}}}{4}$

Step 1.4.1.2.5.4.2

Rewrite the expression.

$\frac{3 \cdot 2^{1}}{4}$

Step 1.4.1.2.5.5

Evaluate the exponent.

$\frac{3 \cdot 2}{4}$

Step 1.4.1.2.6

Multiply $3$ by $2$ .

$\frac{6}{4}$

Step 1.4.1.2.7

Cancel the common factor of $6$ and $4$ .

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Step 1.4.1.2.7.1

Factor $2$ out of $6$ .

$\frac{2 (3)}{4}$

Step 1.4.1.2.7.2

Cancel the common factors.

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Step 1.4.1.2.7.2.1

Factor $2$ out of $4$ .

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

Step 1.4.1.2.7.2.2

Cancel the common factor.

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

Step 1.4.1.2.7.2.3

Rewrite the expression.

$\frac{3}{2}$

Step 1.4.2

Evaluate at $x = \frac{3 π}{4}$ .

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Step 1.4.2.1

Substitute $\frac{3 π}{4}$ for $x$ .

$3 \sin (\frac{3 π}{4}) \cos (\frac{3 π}{4})$

Step 1.4.2.2

Simplify.

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Step 1.4.2.2.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$3 \sin (\frac{π}{4}) \cos (\frac{3 π}{4})$

Step 1.4.2.2.2

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$3 \frac{\sqrt{2}}{2} \cos (\frac{3 π}{4})$

Step 1.4.2.2.3

Combine $3$ and $\frac{\sqrt{2}}{2}$ .

$\frac{3 \sqrt{2}}{2} \cos (\frac{3 π}{4})$

Step 1.4.2.2.4

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$\frac{3 \sqrt{2}}{2} (- \cos (\frac{π}{4}))$

Step 1.4.2.2.5

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$\frac{3 \sqrt{2}}{2} (- \frac{\sqrt{2}}{2})$

Step 1.4.2.2.6

Multiply $\frac{3 \sqrt{2}}{2} (- \frac{\sqrt{2}}{2})$ .

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Step 1.4.2.2.6.1

Multiply $\frac{3 \sqrt{2}}{2}$ by $\frac{\sqrt{2}}{2}$ .

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

Step 1.4.2.2.6.2

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 1.4.2.2.6.3

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 1.4.2.2.6.4

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

$- \frac{3 {\sqrt{2}}^{1 + 1}}{2 \cdot 2}$

Step 1.4.2.2.6.5

Add $1$ and $1$ .

$- \frac{3 {\sqrt{2}}^{2}}{2 \cdot 2}$

Step 1.4.2.2.6.6

Multiply $2$ by $2$ .

$- \frac{3 {\sqrt{2}}^{2}}{4}$

Step 1.4.2.2.7

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 1.4.2.2.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

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

Step 1.4.2.2.7.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 1.4.2.2.7.3

Combine $\frac{1}{2}$ and $2$ .

$- \frac{3 \cdot 2^{\frac{2}{2}}}{4}$

Step 1.4.2.2.7.4

Cancel the common factor of $2$ .

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Step 1.4.2.2.7.4.1

Cancel the common factor.

$- \frac{3 \cdot 2^{\frac{2}{2}}}{4}$

Step 1.4.2.2.7.4.2

Rewrite the expression.

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

Step 1.4.2.2.7.5

Evaluate the exponent.

$- \frac{3 \cdot 2}{4}$

Step 1.4.2.2.8

Multiply $3$ by $2$ .

$- \frac{6}{4}$

Step 1.4.2.2.9

Cancel the common factor of $6$ and $4$ .

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Step 1.4.2.2.9.1

Factor $2$ out of $6$ .

$- \frac{2 (3)}{4}$

Step 1.4.2.2.9.2

Cancel the common factors.

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Step 1.4.2.2.9.2.1

Factor $2$ out of $4$ .

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

Step 1.4.2.2.9.2.2

Cancel the common factor.

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

Step 1.4.2.2.9.2.3

Rewrite the expression.

$- \frac{3}{2}$

Step 1.4.3

List all of the points.

$(\frac{π}{4} + π n, \frac{3}{2}), (\frac{3 π}{4} + π n, - \frac{3}{2})$ , for any integer $n$

Step 2

Exclude the points that are not on the interval.

$(\frac{π}{4}, \frac{3}{2}), (\frac{3 π}{4}, - \frac{3}{2})$

Step 3

Evaluate at the included endpoints.

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Step 3.1

Evaluate at $x = \frac{π}{4}$ .

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Step 3.1.1

Substitute $\frac{π}{4}$ for $x$ .

$3 \sin (\frac{π}{4}) \cos (\frac{π}{4})$

Step 3.1.2

Simplify.

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Step 3.1.2.1

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$3 \frac{\sqrt{2}}{2} \cos (\frac{π}{4})$

Step 3.1.2.2

Combine $3$ and $\frac{\sqrt{2}}{2}$ .

$\frac{3 \sqrt{2}}{2} \cos (\frac{π}{4})$

Step 3.1.2.3

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$\frac{3 \sqrt{2}}{2} \cdot \frac{\sqrt{2}}{2}$

Step 3.1.2.4

Multiply $\frac{3 \sqrt{2}}{2} \cdot \frac{\sqrt{2}}{2}$ .

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Step 3.1.2.4.1

Multiply $\frac{3 \sqrt{2}}{2}$ by $\frac{\sqrt{2}}{2}$ .

$\frac{3 \sqrt{2} \sqrt{2}}{2 \cdot 2}$

Step 3.1.2.4.2

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{3 ({\sqrt{2}}^{1} \sqrt{2})}{2 \cdot 2}$

Step 3.1.2.4.3

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{3 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}{2 \cdot 2}$

Step 3.1.2.4.4

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

$\frac{3 {\sqrt{2}}^{1 + 1}}{2 \cdot 2}$

Step 3.1.2.4.5

Add $1$ and $1$ .

$\frac{3 {\sqrt{2}}^{2}}{2 \cdot 2}$

Step 3.1.2.4.6

Multiply $2$ by $2$ .

$\frac{3 {\sqrt{2}}^{2}}{4}$

Step 3.1.2.5

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 3.1.2.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

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

Step 3.1.2.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{3 \cdot 2^{\frac{1}{2} \cdot 2}}{4}$

Step 3.1.2.5.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{3 \cdot 2^{\frac{2}{2}}}{4}$

Step 3.1.2.5.4

Cancel the common factor of $2$ .

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Step 3.1.2.5.4.1

Cancel the common factor.

$\frac{3 \cdot 2^{\frac{2}{2}}}{4}$

Step 3.1.2.5.4.2

Rewrite the expression.

$\frac{3 \cdot 2^{1}}{4}$

Step 3.1.2.5.5

Evaluate the exponent.

$\frac{3 \cdot 2}{4}$

Step 3.1.2.6

Multiply $3$ by $2$ .

$\frac{6}{4}$

Step 3.1.2.7

Cancel the common factor of $6$ and $4$ .

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Step 3.1.2.7.1

Factor $2$ out of $6$ .

$\frac{2 (3)}{4}$

Step 3.1.2.7.2

Cancel the common factors.

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Step 3.1.2.7.2.1

Factor $2$ out of $4$ .

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

Step 3.1.2.7.2.2

Cancel the common factor.

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

Step 3.1.2.7.2.3

Rewrite the expression.

$\frac{3}{2}$

Step 3.2

Evaluate at $x = π$ .

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Step 3.2.1

Substitute $π$ for $x$ .

$3 \sin (π) \cos (π)$

Step 3.2.2

Simplify.

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Step 3.2.2.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$3 \sin (0) \cos (π)$

Step 3.2.2.2

The exact value of $\sin (0)$ is $0$ .

$3 \cdot 0 \cos (π)$

Step 3.2.2.3

Multiply $3$ by $0$ .

$0 \cos (π)$

Step 3.2.2.4

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$0 (- \cos (0))$

Step 3.2.2.5

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

$0 (- 1 \cdot 1)$

Step 3.2.2.6

Multiply $0 (- 1 \cdot 1)$ .

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Step 3.2.2.6.1

Multiply $- 1$ by $1$ .

$0 \cdot - 1$

Step 3.2.2.6.2

Multiply $0$ by $- 1$ .

$0$

Step 3.3

List all of the points.

$(\frac{π}{4}, \frac{3}{2}), (π, 0)$

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: $(\frac{π}{4}, \frac{3}{2})$

Absolute Minimum: $(\frac{3 π}{4}, - \frac{3}{2})$

Step 5