Calculus Examples

$y = 8 \cos (\frac{π}{4} x - \frac{π}{2})$

Step 1

Find the first derivative of the function.

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Step 1.1

Differentiate using the Constant Multiple Rule.

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Step 1.1.1

Combine $\frac{π}{4}$ and $x$ .

$\frac{d}{d x} [8 \cos (\frac{π x}{4} - \frac{π}{2})]$

Step 1.1.2

Since $8$ is constant with respect to $x$ , the derivative of $8 \cos (\frac{π x}{4} - \frac{π}{2})$ with respect to $x$ is $8 \frac{d}{d x} [\cos (\frac{π x}{4} - \frac{π}{2})]$ .

$8 \frac{d}{d x} [\cos (\frac{π x}{4} - \frac{π}{2})]$

Step 1.2

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 $g (x) = \frac{π x}{4} - \frac{π}{2}$ .

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Step 1.2.1

To apply the Chain Rule, set $u$ as $\frac{π x}{4} - \frac{π}{2}$ .

$8 (\frac{d}{d u} [\cos (u)] \frac{d}{d x} [\frac{π x}{4} - \frac{π}{2}])$

Step 1.2.2

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

$8 (- \sin (u) \frac{d}{d x} [\frac{π x}{4} - \frac{π}{2}])$

Step 1.2.3

Replace all occurrences of $u$ with $\frac{π x}{4} - \frac{π}{2}$ .

$8 (- \sin (\frac{π x}{4} - \frac{π}{2}) \frac{d}{d x} [\frac{π x}{4} - \frac{π}{2}])$

Step 1.3

Differentiate.

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Step 1.3.1

Multiply $- 1$ by $8$ .

$- 8 (\sin (\frac{π x}{4} - \frac{π}{2}) \frac{d}{d x} [\frac{π x}{4} - \frac{π}{2}])$

Step 1.3.2

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

$- 8 \sin (\frac{π x}{4} - \frac{π}{2}) (\frac{d}{d x} [\frac{π x}{4}] + \frac{d}{d x} [- \frac{π}{2}])$

Step 1.3.3

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

$- 8 \sin (\frac{π x}{4} - \frac{π}{2}) (\frac{π}{4} \frac{d}{d x} [x] + \frac{d}{d x} [- \frac{π}{2}])$

Step 1.3.4

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

$- 8 \sin (\frac{π x}{4} - \frac{π}{2}) (\frac{π}{4} \cdot 1 + \frac{d}{d x} [- \frac{π}{2}])$

Step 1.3.5

Multiply $\frac{π}{4}$ by $1$ .

$- 8 \sin (\frac{π x}{4} - \frac{π}{2}) (\frac{π}{4} + \frac{d}{d x} [- \frac{π}{2}])$

Step 1.3.6

Since $- \frac{π}{2}$ is constant with respect to $x$ , the derivative of $- \frac{π}{2}$ with respect to $x$ is $0$ .

$- 8 \sin (\frac{π x}{4} - \frac{π}{2}) (\frac{π}{4} + 0)$

Step 1.3.7

Simplify terms.

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Step 1.3.7.1

Add $\frac{π}{4}$ and $0$ .

$- 8 \sin (\frac{π x}{4} - \frac{π}{2}) \frac{π}{4}$

Step 1.3.7.2

Combine $\frac{π}{4}$ and $- 8$ .

$\frac{π \cdot - 8}{4} \sin (\frac{π x}{4} - \frac{π}{2})$

Step 1.3.7.3

Combine $\frac{π \cdot - 8}{4}$ and $\sin (\frac{π x}{4} - \frac{π}{2})$ .

$\frac{π \cdot - 8 \sin (\frac{π x}{4} - \frac{π}{2})}{4}$

Step 1.3.7.4

Move $- 8$ to the left of $π$ .

$\frac{- 8 \cdot π \sin (\frac{π x}{4} - \frac{π}{2})}{4}$

Step 1.3.7.5

Cancel the common factor of $- 8$ and $4$ .

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Step 1.3.7.5.1

Factor $4$ out of $- 8 π \sin (\frac{π x}{4} - \frac{π}{2})$ .

$\frac{4 (- 2 π \sin (\frac{π x}{4} - \frac{π}{2}))}{4}$

Step 1.3.7.5.2

Cancel the common factors.

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Step 1.3.7.5.2.1

Factor $4$ out of $4$ .

$\frac{4 (- 2 π \sin (\frac{π x}{4} - \frac{π}{2}))}{4 (1)}$

Step 1.3.7.5.2.2

Cancel the common factor.

$\frac{4 (- 2 π \sin (\frac{π x}{4} - \frac{π}{2}))}{4 \cdot 1}$

Step 1.3.7.5.2.3

Rewrite the expression.

$\frac{- 2 π \sin (\frac{π x}{4} - \frac{π}{2})}{1}$

Step 1.3.7.5.2.4

Divide $- 2 π \sin (\frac{π x}{4} - \frac{π}{2})$ by $1$ .

$- 2 π \sin (\frac{π x}{4} - \frac{π}{2})$

Step 2

Find the second derivative of the function.

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Step 2.1

Since $- 2 π$ is constant with respect to $x$ , the derivative of $- 2 π \sin (\frac{π x}{4} - \frac{π}{2})$ with respect to $x$ is $- 2 π \frac{d}{d x} [\sin (\frac{π x}{4} - \frac{π}{2})]$ .

$f'' (x) = - 2 π \frac{d}{d x} \sin (\frac{π x}{4} - \frac{π}{2})$

Step 2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = \frac{π x}{4} - \frac{π}{2}$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $\frac{π x}{4} - \frac{π}{2}$ .

$f'' (x) = - 2 π (\frac{d}{d u} (\sin (u)) \frac{d}{d x} (\frac{π x}{4} - \frac{π}{2}))$

Step 2.2.2

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

$f'' (x) = - 2 π (\cos (u) \frac{d}{d x} (\frac{π x}{4} - \frac{π}{2}))$

Step 2.2.3

Replace all occurrences of $u$ with $\frac{π x}{4} - \frac{π}{2}$ .

$f'' (x) = - 2 π (\cos (\frac{π x}{4} - \frac{π}{2}) \frac{d}{d x} (\frac{π x}{4} - \frac{π}{2}))$

Step 2.3

Differentiate.

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Step 2.3.1

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

$f'' (x) = - 2 π \cos (\frac{π x}{4} - \frac{π}{2}) (\frac{d}{d x} (\frac{π x}{4}) + \frac{d}{d x} (- \frac{π}{2}))$

Step 2.3.2

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

$f'' (x) = - 2 π \cos (\frac{π x}{4} - \frac{π}{2}) (\frac{π}{4} \cdot \frac{d}{d x} (x) + \frac{d}{d x} (- \frac{π}{2}))$

Step 2.3.3

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

$f'' (x) = - 2 π \cos (\frac{π x}{4} - \frac{π}{2}) (\frac{π}{4} \cdot 1 + \frac{d}{d x} (- \frac{π}{2}))$

Step 2.3.4

Multiply $\frac{π}{4}$ by $1$ .

$f'' (x) = - 2 π \cos (\frac{π x}{4} - \frac{π}{2}) (\frac{π}{4} + \frac{d}{d x} (- \frac{π}{2}))$

Step 2.3.5

Since $- \frac{π}{2}$ is constant with respect to $x$ , the derivative of $- \frac{π}{2}$ with respect to $x$ is $0$ .

$f'' (x) = - 2 π \cos (\frac{π x}{4} - \frac{π}{2}) (\frac{π}{4} + 0)$

Step 2.3.6

Combine fractions.

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Step 2.3.6.1

Add $\frac{π}{4}$ and $0$ .

$f'' (x) = - 2 π \cos (\frac{π x}{4} - \frac{π}{2}) \frac{π}{4}$

Step 2.3.6.2

Combine $\frac{π}{4}$ and $- 2$ .

$f'' (x) = \frac{π \cdot - 2}{4} \cdot (π \cos (\frac{π x}{4} - \frac{π}{2}))$

Step 2.3.6.3

Combine $\frac{π \cdot - 2}{4}$ and $π$ .

$f'' (x) = \frac{π \cdot (- 2 π)}{4} \cdot \cos (\frac{π x}{4} - \frac{π}{2})$

Step 2.4

Raise $π$ to the power of $1$ .

$f'' (x) = \frac{- 2 (π π)}{4} \cdot \cos (\frac{π x}{4} - \frac{π}{2})$

Step 2.5

Raise $π$ to the power of $1$ .

$f'' (x) = \frac{- 2 (π π)}{4} \cdot \cos (\frac{π x}{4} - \frac{π}{2})$

Step 2.6

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

$f'' (x) = \frac{- 2 π^{1 + 1}}{4} \cdot \cos (\frac{π x}{4} - \frac{π}{2})$

Step 2.7

Add $1$ and $1$ .

$f'' (x) = \frac{- 2 π^{2}}{4} \cdot \cos (\frac{π x}{4} - \frac{π}{2})$

Step 2.8

Combine $\frac{- 2 π^{2}}{4}$ and $\cos (\frac{π x}{4} - \frac{π}{2})$ .

$f'' (x) = \frac{- 2 π^{2} \cos (\frac{π x}{4} - \frac{π}{2})}{4}$

Step 2.9

Cancel the common factor of $- 2$ and $4$ .

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Step 2.9.1

Factor $2$ out of $- 2 π^{2} \cos (\frac{π x}{4} - \frac{π}{2})$ .

$f'' (x) = \frac{2 (- π^{2} \cos (\frac{π x}{4} - \frac{π}{2}))}{4}$

Step 2.9.2

Cancel the common factors.

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Step 2.9.2.1

Factor $2$ out of $4$ .

$f'' (x) = \frac{2 (- π^{2} \cos (\frac{π x}{4} - \frac{π}{2}))}{2 (2)}$

Step 2.9.2.2

Cancel the common factor.

$f'' (x) = \frac{2 (- π^{2} \cos (\frac{π x}{4} - \frac{π}{2}))}{2 \cdot 2}$

Step 2.9.2.3

Rewrite the expression.

$f'' (x) = \frac{- π^{2} \cos (\frac{π x}{4} - \frac{π}{2})}{2}$

Step 2.10

Move the negative in front of the fraction.

$f'' (x) = - \frac{π^{2} \cos (\frac{π x}{4} - \frac{π}{2})}{2}$

Step 3

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$- 2 π \sin (\frac{π x}{4} - \frac{π}{2}) = 0$

Step 4

Divide each term in $- 2 π \sin (\frac{π x}{4} - \frac{π}{2}) = 0$ by $- 2 π$ and simplify.

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Step 4.1

Divide each term in $- 2 π \sin (\frac{π x}{4} - \frac{π}{2}) = 0$ by $- 2 π$ .

$\frac{- 2 π \sin (\frac{π x}{4} - \frac{π}{2})}{- 2 π} = \frac{0}{- 2 π}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $- 2$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{- 2 π \sin (\frac{π x}{4} - \frac{π}{2})}{- 2 π} = \frac{0}{- 2 π}$

Step 4.2.1.2

Rewrite the expression.

$\frac{π \sin (\frac{π x}{4} - \frac{π}{2})}{π} = \frac{0}{- 2 π}$

Step 4.2.2

Cancel the common factor of $π$ .

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Step 4.2.2.1

Cancel the common factor.

$\frac{π \sin (\frac{π x}{4} - \frac{π}{2})}{π} = \frac{0}{- 2 π}$

Step 4.2.2.2

Divide $\sin (\frac{π x}{4} - \frac{π}{2})$ by $1$ .

$\sin (\frac{π x}{4} - \frac{π}{2}) = \frac{0}{- 2 π}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Cancel the common factor of $0$ and $- 2$ .

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Step 4.3.1.1

Factor $2$ out of $0$ .

$\sin (\frac{π x}{4} - \frac{π}{2}) = \frac{2 (0)}{- 2 π}$

Step 4.3.1.2

Cancel the common factors.

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Step 4.3.1.2.1

Factor $2$ out of $- 2 π$ .

$\sin (\frac{π x}{4} - \frac{π}{2}) = \frac{2 (0)}{2 (- π)}$

Step 4.3.1.2.2

Cancel the common factor.

$\sin (\frac{π x}{4} - \frac{π}{2}) = \frac{2 \cdot 0}{2 (- π)}$

Step 4.3.1.2.3

Rewrite the expression.

$\sin (\frac{π x}{4} - \frac{π}{2}) = \frac{0}{- π}$

Step 4.3.2

Divide $0$ by $- π$ .

$\sin (\frac{π x}{4} - \frac{π}{2}) = 0$

Step 5

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

$\frac{π x}{4} - \frac{π}{2} = \arcsin (0)$

Step 6

Simplify the right side.

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Step 6.1

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

$\frac{π x}{4} - \frac{π}{2} = 0$

Step 7

Add $\frac{π}{2}$ to both sides of the equation.

$\frac{π x}{4} = \frac{π}{2}$

Step 8

Multiply both sides of the equation by $\frac{4}{π}$ .

$\frac{4}{π} \cdot \frac{π x}{4} = \frac{4}{π} \cdot \frac{π}{2}$

Step 9

Simplify both sides of the equation.

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Step 9.1

Simplify the left side.

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Step 9.1.1

Simplify $\frac{4}{π} \cdot \frac{π x}{4}$ .

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Step 9.1.1.1

Cancel the common factor of $4$ .

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Step 9.1.1.1.1

Cancel the common factor.

$\frac{4}{π} \cdot \frac{π x}{4} = \frac{4}{π} \cdot \frac{π}{2}$

Step 9.1.1.1.2

Rewrite the expression.

$\frac{1}{π} (π x) = \frac{4}{π} \cdot \frac{π}{2}$

Step 9.1.1.2

Cancel the common factor of $π$ .

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Step 9.1.1.2.1

Factor $π$ out of $π x$ .

$\frac{1}{π} (π (x)) = \frac{4}{π} \cdot \frac{π}{2}$

Step 9.1.1.2.2

Cancel the common factor.

$\frac{1}{π} (π x) = \frac{4}{π} \cdot \frac{π}{2}$

Step 9.1.1.2.3

Rewrite the expression.

$x = \frac{4}{π} \cdot \frac{π}{2}$

Step 9.2

Simplify the right side.

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Step 9.2.1

Simplify $\frac{4}{π} \cdot \frac{π}{2}$ .

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Step 9.2.1.1

Cancel the common factor of $2$ .

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Step 9.2.1.1.1

Factor $2$ out of $4$ .

$x = \frac{2 (2)}{π} \cdot \frac{π}{2}$

Step 9.2.1.1.2

Cancel the common factor.

$x = \frac{2 \cdot 2}{π} \cdot \frac{π}{2}$

Step 9.2.1.1.3

Rewrite the expression.

$x = \frac{2}{π} π$

Step 9.2.1.2

Cancel the common factor of $π$ .

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Step 9.2.1.2.1

Cancel the common factor.

$x = \frac{2}{π} π$

Step 9.2.1.2.2

Rewrite the expression.

$x = 2$

Step 10

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from $π$ to find the solution in the second quadrant.

$\frac{π x}{4} - \frac{π}{2} = π - 0$

Step 11

Solve for $x$ .

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Step 11.1

Subtract $0$ from $π$ .

$\frac{π x}{4} - \frac{π}{2} = π$

Step 11.2

Move all terms not containing $x$ to the right side of the equation.

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Step 11.2.1

Add $\frac{π}{2}$ to both sides of the equation.

$\frac{π x}{4} = π + \frac{π}{2}$

Step 11.2.2

To write $π$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{π x}{4} = π \cdot \frac{2}{2} + \frac{π}{2}$

Step 11.2.3

Combine $π$ and $\frac{2}{2}$ .

$\frac{π x}{4} = \frac{π \cdot 2}{2} + \frac{π}{2}$

Step 11.2.4

Combine the numerators over the common denominator.

$\frac{π x}{4} = \frac{π \cdot 2 + π}{2}$

Step 11.2.5

Simplify the numerator.

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Step 11.2.5.1

Move $2$ to the left of $π$ .

$\frac{π x}{4} = \frac{2 \cdot π + π}{2}$

Step 11.2.5.2

Add $2 π$ and $π$ .

$\frac{π x}{4} = \frac{3 π}{2}$

Step 11.3

Multiply both sides of the equation by $\frac{4}{π}$ .

$\frac{4}{π} \cdot \frac{π x}{4} = \frac{4}{π} \cdot \frac{3 π}{2}$

Step 11.4

Simplify both sides of the equation.

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Step 11.4.1

Simplify the left side.

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Step 11.4.1.1

Simplify $\frac{4}{π} \cdot \frac{π x}{4}$ .

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Step 11.4.1.1.1

Cancel the common factor of $4$ .

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Step 11.4.1.1.1.1

Cancel the common factor.

$\frac{4}{π} \cdot \frac{π x}{4} = \frac{4}{π} \cdot \frac{3 π}{2}$

Step 11.4.1.1.1.2

Rewrite the expression.

$\frac{1}{π} (π x) = \frac{4}{π} \cdot \frac{3 π}{2}$

Step 11.4.1.1.2

Cancel the common factor of $π$ .

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Step 11.4.1.1.2.1

Factor $π$ out of $π x$ .

$\frac{1}{π} (π (x)) = \frac{4}{π} \cdot \frac{3 π}{2}$

Step 11.4.1.1.2.2

Cancel the common factor.

$\frac{1}{π} (π x) = \frac{4}{π} \cdot \frac{3 π}{2}$

Step 11.4.1.1.2.3

Rewrite the expression.

$x = \frac{4}{π} \cdot \frac{3 π}{2}$

Step 11.4.2

Simplify the right side.

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Step 11.4.2.1

Simplify $\frac{4}{π} \cdot \frac{3 π}{2}$ .

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Step 11.4.2.1.1

Cancel the common factor of $2$ .

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Step 11.4.2.1.1.1

Factor $2$ out of $4$ .

$x = \frac{2 (2)}{π} \cdot \frac{3 π}{2}$

Step 11.4.2.1.1.2

Cancel the common factor.

$x = \frac{2 \cdot 2}{π} \cdot \frac{3 π}{2}$

Step 11.4.2.1.1.3

Rewrite the expression.

$x = \frac{2}{π} (3 π)$

Step 11.4.2.1.2

Cancel the common factor of $π$ .

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Step 11.4.2.1.2.1

Factor $π$ out of $3 π$ .

$x = \frac{2}{π} (π \cdot 3)$

Step 11.4.2.1.2.2

Cancel the common factor.

$x = \frac{2}{π} (π \cdot 3)$

Step 11.4.2.1.2.3

Rewrite the expression.

$x = 2 \cdot 3$

Step 11.4.2.1.3

Multiply $2$ by $3$ .

$x = 6$

Step 12

The solution to the equation $\frac{π x}{4} - \frac{π}{2} = 0$ .

$x = 2, 6$

Step 13

Evaluate the second derivative at $x = 2$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

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

Step 14

Evaluate the second derivative.

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Step 14.1

Cancel the common factor of $2$ and $4$ .

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Step 14.1.1

Factor $2$ out of $π (2)$ .

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

Step 14.1.2

Cancel the common factors.

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Step 14.1.2.1

Factor $2$ out of $4$ .

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

Step 14.1.2.2

Cancel the common factor.

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

Step 14.1.2.3

Rewrite the expression.

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

Step 14.2

Simplify the numerator.

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Step 14.2.1

Combine the numerators over the common denominator.

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

Step 14.2.2

Subtract $π$ from $π$ .

$- \frac{π^{2} \cos (\frac{0}{2})}{2}$

Step 14.2.3

Divide $0$ by $2$ .

$- \frac{π^{2} \cos (0)}{2}$

Step 14.2.4

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

$- \frac{π^{2} \cdot 1}{2}$

Step 14.3

Multiply $π^{2}$ by $1$ .

$- \frac{π^{2}}{2}$

Step 15

$x = 2$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = 2$ is a local maximum

Step 16

Find the y-value when $x = 2$ .

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Step 16.1

Replace the variable $x$ with $2$ in the expression.

$f (2) = 8 \cos (\frac{π}{4} \cdot (2) - \frac{π}{2})$

Step 16.2

Simplify the result.

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Step 16.2.1

Cancel the common factor of $2$ .

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Step 16.2.1.1

Factor $2$ out of $4$ .

$f (2) = 8 \cos (\frac{π}{2 (2)} \cdot 2 - \frac{π}{2})$

Step 16.2.1.2

Cancel the common factor.

$f (2) = 8 \cos (\frac{π}{2 \cdot 2} \cdot 2 - \frac{π}{2})$

Step 16.2.1.3

Rewrite the expression.

$f (2) = 8 \cos (\frac{π}{2} - \frac{π}{2})$

Step 16.2.2

Combine the numerators over the common denominator.

$f (2) = 8 \cos (\frac{π - π}{2})$

Step 16.2.3

Subtract $π$ from $π$ .

$f (2) = 8 \cos (\frac{0}{2})$

Step 16.2.4

Divide $0$ by $2$ .

$f (2) = 8 \cos (0)$

Step 16.2.5

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

$f (2) = 8 \cdot 1$

Step 16.2.6

Multiply $8$ by $1$ .

$f (2) = 8$

Step 16.2.7

The final answer is $8$ .

$y = 8$

Step 17

Evaluate the second derivative at $x = 6$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- \frac{π^{2} \cos (\frac{π (6)}{4} - \frac{π}{2})}{2}$

Step 18

Evaluate the second derivative.

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Step 18.1

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

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Step 18.1.1

Factor $2$ out of $π (6)$ .

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

Step 18.1.2

Cancel the common factors.

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Step 18.1.2.1

Factor $2$ out of $4$ .

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

Step 18.1.2.2

Cancel the common factor.

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

Step 18.1.2.3

Rewrite the expression.

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

Step 18.2

Simplify the numerator.

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Step 18.2.1

Combine the numerators over the common denominator.

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

Step 18.2.2

Move $3$ to the left of $π$ .

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

Step 18.2.3

Subtract $π$ from $3 π$ .

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

Step 18.2.4

Cancel the common factor of $2$ .

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Step 18.2.4.1

Cancel the common factor.

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

Step 18.2.4.2

Divide $π$ by $1$ .

$- \frac{π^{2} \cos (π)}{2}$

Step 18.2.5

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.

$- \frac{π^{2} (- \cos (0))}{2}$

Step 18.2.6

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

$- \frac{π^{2} (- 1 \cdot 1)}{2}$

Step 18.2.7

Multiply $- 1$ by $1$ .

$- \frac{π^{2} \cdot - 1}{2}$

Step 18.3

Simplify the expression.

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Step 18.3.1

Move $- 1$ to the left of $π^{2}$ .

$- \frac{- 1 \cdot π^{2}}{2}$

Step 18.3.2

Move the negative in front of the fraction.

$- - \frac{π^{2}}{2}$

Step 18.4

Multiply $- - \frac{π^{2}}{2}$ .

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Step 18.4.1

Multiply $- 1$ by $- 1$ .

$1 \frac{π^{2}}{2}$

Step 18.4.2

Multiply $\frac{π^{2}}{2}$ by $1$ .

$\frac{π^{2}}{2}$

Step 19

$x = 6$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = 6$ is a local minimum

Step 20

Find the y-value when $x = 6$ .

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Step 20.1

Replace the variable $x$ with $6$ in the expression.

$f (6) = 8 \cos (\frac{π}{4} \cdot (6) - \frac{π}{2})$

Step 20.2

Simplify the result.

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Step 20.2.1

Simplify each term.

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Step 20.2.1.1

Cancel the common factor of $2$ .

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Step 20.2.1.1.1

Factor $2$ out of $4$ .

$f (6) = 8 \cos (\frac{π}{2 (2)} \cdot 6 - \frac{π}{2})$

Step 20.2.1.1.2

Factor $2$ out of $6$ .

$f (6) = 8 \cos (\frac{π}{2 \cdot 2} \cdot (2 \cdot 3) - \frac{π}{2})$

Step 20.2.1.1.3

Cancel the common factor.

$f (6) = 8 \cos (\frac{π}{2 \cdot 2} \cdot (2 \cdot 3) - \frac{π}{2})$

Step 20.2.1.1.4

Rewrite the expression.

$f (6) = 8 \cos (\frac{π}{2} \cdot 3 - \frac{π}{2})$

Step 20.2.1.2

Combine $\frac{π}{2}$ and $3$ .

$f (6) = 8 \cos (\frac{π \cdot 3}{2} - \frac{π}{2})$

Step 20.2.1.3

Move $3$ to the left of $π$ .

$f (6) = 8 \cos (\frac{3 π}{2} - \frac{π}{2})$

Step 20.2.2

Simplify terms.

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Step 20.2.2.1

Combine the numerators over the common denominator.

$f (6) = 8 \cos (\frac{3 π - π}{2})$

Step 20.2.2.2

Subtract $π$ from $3 π$ .

$f (6) = 8 \cos (\frac{2 π}{2})$

Step 20.2.2.3

Cancel the common factor of $2$ .

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Step 20.2.2.3.1

Cancel the common factor.

$f (6) = 8 \cos (\frac{2 π}{2})$

Step 20.2.2.3.2

Divide $π$ by $1$ .

$f (6) = 8 \cos (π)$

Step 20.2.3

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.

$f (6) = 8 (- \cos (0))$

Step 20.2.4

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

$f (6) = 8 (- 1 \cdot 1)$

Step 20.2.5

Multiply $8 (- 1 \cdot 1)$ .

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Step 20.2.5.1

Multiply $- 1$ by $1$ .

$f (6) = 8 \cdot - 1$

Step 20.2.5.2

Multiply $8$ by $- 1$ .

$f (6) = - 8$

Step 20.2.6

The final answer is $- 8$ .

$y = - 8$

Step 21

These are the local extrema for $f (x) = 8 \cos (\frac{π}{4} x - \frac{π}{2})$ .

$(2, 8)$ is a local maxima

$(6, - 8)$ is a local minima

Step 22