Calculus Examples

$y = 10 \cos (\frac{2 π}{3} (x + \frac{1}{4}))$

Step 1

Find the first derivative of the function.

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Step 1.1

Since $10$ is constant with respect to $x$ , the derivative of $10 \cos (\frac{2 π}{3} (x + \frac{1}{4}))$ with respect to $x$ is $10 \frac{d}{d x} [\cos (\frac{2 π}{3} (x + \frac{1}{4}))]$ .

$10 \frac{d}{d x} [\cos (\frac{2 π}{3} (x + \frac{1}{4}))]$

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 $g (x) = \frac{2 π}{3} (x + \frac{1}{4})$ .

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Step 1.2.1

To apply the Chain Rule, set $u$ as $\frac{2 π}{3} (x + \frac{1}{4})$ .

$10 (\frac{d}{d u} [\cos (u)] \frac{d}{d x} [\frac{2 π}{3} (x + \frac{1}{4})])$

Step 1.2.2

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

$10 (- \sin (\frac{2 π}{3} (x + \frac{1}{4})) \frac{d}{d x} [\frac{2 π}{3} (x + \frac{1}{4})])$

Step 1.3

Differentiate.

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Step 1.3.1

Multiply $- 1$ by $10$ .

$- 10 (\sin (\frac{2 π}{3} (x + \frac{1}{4})) \frac{d}{d x} [\frac{2 π}{3} (x + \frac{1}{4})])$

Step 1.3.2

Since $\frac{2 π}{3}$ is constant with respect to $x$ , the derivative of $\frac{2 π}{3} (x + \frac{1}{4})$ with respect to $x$ is $\frac{2 π}{3} \frac{d}{d x} [x + \frac{1}{4}]$ .

$- 10 \sin (\frac{2 π}{3} (x + \frac{1}{4})) (\frac{2 π}{3} \frac{d}{d x} [x + \frac{1}{4}])$

Step 1.3.3

Combine fractions.

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Step 1.3.3.1

Combine $\frac{2 π}{3}$ and $- 10$ .

$\frac{2 π \cdot - 10}{3} \sin (\frac{2 π}{3} (x + \frac{1}{4})) \frac{d}{d x} [x + \frac{1}{4}]$

Step 1.3.3.2

Multiply $- 10$ by $2$ .

$\frac{- 20 π}{3} \sin (\frac{2 π}{3} (x + \frac{1}{4})) \frac{d}{d x} [x + \frac{1}{4}]$

Step 1.3.3.3

Combine $\frac{- 20 π}{3}$ and $\sin (\frac{2 π}{3} (x + \frac{1}{4}))$ .

$\frac{- 20 π \sin (\frac{2 π}{3} (x + \frac{1}{4}))}{3} \frac{d}{d x} [x + \frac{1}{4}]$

Step 1.3.3.4

Move the negative in front of the fraction.

$- \frac{20 π \sin (\frac{2 π}{3} (x + \frac{1}{4}))}{3} \frac{d}{d x} [x + \frac{1}{4}]$

Step 1.3.4

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

$- \frac{20 π \sin (\frac{2 π}{3} (x + \frac{1}{4}))}{3} (\frac{d}{d x} [x] + \frac{d}{d x} [\frac{1}{4}])$

Step 1.3.5

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

$- \frac{20 π \sin (\frac{2 π}{3} (x + \frac{1}{4}))}{3} (1 + \frac{d}{d x} [\frac{1}{4}])$

Step 1.3.6

Since $\frac{1}{4}$ is constant with respect to $x$ , the derivative of $\frac{1}{4}$ with respect to $x$ is $0$ .

$- \frac{20 π \sin (\frac{2 π}{3} (x + \frac{1}{4}))}{3} (1 + 0)$

Step 1.3.7

Simplify the expression.

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Step 1.3.7.1

Add $1$ and $0$ .

$- \frac{20 π \sin (\frac{2 π}{3} (x + \frac{1}{4}))}{3} \cdot 1$

Step 1.3.7.2

Multiply $- 1$ by $1$ .

$- \frac{20 π \sin (\frac{2 π}{3} (x + \frac{1}{4}))}{3}$

Step 1.4

Simplify.

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Step 1.4.1

Apply the distributive property.

$- \frac{20 π \sin (\frac{2 π}{3} x + \frac{2 π}{3} \cdot \frac{1}{4})}{3}$

Step 1.4.2

Simplify each term.

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Step 1.4.2.1

Combine $\frac{2 π}{3}$ and $x$ .

$- \frac{20 π \sin (\frac{2 π x}{3} + \frac{2 π}{3} \cdot \frac{1}{4})}{3}$

Step 1.4.2.2

Cancel the common factor of $2$ .

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Step 1.4.2.2.1

Factor $2$ out of $2 π$ .

$- \frac{20 π \sin (\frac{2 π x}{3} + \frac{2 (π)}{3} \cdot \frac{1}{4})}{3}$

Step 1.4.2.2.2

Factor $2$ out of $4$ .

$- \frac{20 π \sin (\frac{2 π x}{3} + \frac{2 (π)}{3} \cdot \frac{1}{2 (2)})}{3}$

Step 1.4.2.2.3

Cancel the common factor.

$- \frac{20 π \sin (\frac{2 π x}{3} + \frac{2 π}{3} \cdot \frac{1}{2 \cdot 2})}{3}$

Step 1.4.2.2.4

Rewrite the expression.

$- \frac{20 π \sin (\frac{2 π x}{3} + \frac{π}{3} \cdot \frac{1}{2})}{3}$

Step 1.4.2.3

Multiply $\frac{π}{3}$ by $\frac{1}{2}$ .

$- \frac{20 π \sin (\frac{2 π x}{3} + \frac{π}{3 \cdot 2})}{3}$

Step 1.4.2.4

Multiply $3$ by $2$ .

$- \frac{20 π \sin (\frac{2 π x}{3} + \frac{π}{6})}{3}$

Step 2

Find the second derivative of the function.

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Step 2.1

Since $- \frac{20 π}{3}$ is constant with respect to $x$ , the derivative of $- \frac{20 π \sin (\frac{2 π x}{3} + \frac{π}{6})}{3}$ with respect to $x$ is $- \frac{20 π}{3} \frac{d}{d x} [\sin (\frac{2 π x}{3} + \frac{π}{6})]$ .

$f'' (x) = - \frac{20 π}{3} \cdot \frac{d}{d x} \sin (\frac{2 π x}{3} + \frac{π}{6})$

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 $g (x) = \frac{2 π x}{3} + \frac{π}{6}$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $\frac{2 π x}{3} + \frac{π}{6}$ .

$f'' (x) = - \frac{20 π}{3} \cdot (\frac{d}{d u} (\sin (u)) \frac{d}{d x} (\frac{2 π x}{3} + \frac{π}{6}))$

Step 2.2.2

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

$f'' (x) = - \frac{20 π}{3} \cdot (\cos (u) \frac{d}{d x} (\frac{2 π x}{3} + \frac{π}{6}))$

Step 2.2.3

Replace all occurrences of $u$ with $\frac{2 π x}{3} + \frac{π}{6}$ .

$f'' (x) = - \frac{20 π}{3} \cdot (\cos (\frac{2 π x}{3} + \frac{π}{6}) \frac{d}{d x} (\frac{2 π x}{3} + \frac{π}{6}))$

Step 2.3

Differentiate.

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Step 2.3.1

Combine $\cos (\frac{2 π x}{3} + \frac{π}{6})$ and $\frac{20 π}{3}$ .

$f'' (x) = - \frac{\cos (\frac{2 π x}{3} + \frac{π}{6}) (20 π)}{3} \cdot \frac{d}{d x} (\frac{2 π x}{3} + \frac{π}{6})$

Step 2.3.2

Move $20$ to the left of $\cos (\frac{2 π x}{3} + \frac{π}{6})$ .

$f'' (x) = - \frac{20 \cdot (\cos (\frac{2 π x}{3} + \frac{π}{6}) π)}{3} \cdot \frac{d}{d x} (\frac{2 π x}{3} + \frac{π}{6})$

Step 2.3.3

By the Sum Rule, the derivative of $\frac{2 π x}{3} + \frac{π}{6}$ with respect to $x$ is $\frac{d}{d x} [\frac{2 π x}{3}] + \frac{d}{d x} [\frac{π}{6}]$ .

$f'' (x) = - \frac{20 \cos (\frac{2 π x}{3} + \frac{π}{6}) π}{3} \cdot (\frac{d}{d x} (\frac{2 π x}{3}) + \frac{d}{d x} (\frac{π}{6}))$

Step 2.3.4

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

$f'' (x) = - \frac{20 \cos (\frac{2 π x}{3} + \frac{π}{6}) π}{3} \cdot (\frac{2 π}{3} \cdot \frac{d}{d x} (x) + \frac{d}{d x} (\frac{π}{6}))$

Step 2.3.5

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

$f'' (x) = - \frac{20 \cos (\frac{2 π x}{3} + \frac{π}{6}) π}{3} \cdot (\frac{2 π}{3} \cdot 1 + \frac{d}{d x} (\frac{π}{6}))$

Step 2.3.6

Multiply $\frac{2 π}{3}$ by $1$ .

$f'' (x) = - \frac{20 \cos (\frac{2 π x}{3} + \frac{π}{6}) π}{3} \cdot (\frac{2 π}{3} + \frac{d}{d x} (\frac{π}{6}))$

Step 2.3.7

Since $\frac{π}{6}$ is constant with respect to $x$ , the derivative of $\frac{π}{6}$ with respect to $x$ is $0$ .

$f'' (x) = - \frac{20 \cos (\frac{2 π x}{3} + \frac{π}{6}) π}{3} \cdot (\frac{2 π}{3} + 0)$

Step 2.3.8

Combine fractions.

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Step 2.3.8.1

Add $\frac{2 π}{3}$ and $0$ .

$f'' (x) = - \frac{20 \cos (\frac{2 π x}{3} + \frac{π}{6}) π}{3} \cdot \frac{2 π}{3}$

Step 2.3.8.2

Multiply $\frac{2 π}{3}$ by $\frac{20 \cos (\frac{2 π x}{3} + \frac{π}{6}) π}{3}$ .

$f'' (x) = - \frac{2 π (20 \cos (\frac{2 π x}{3} + \frac{π}{6}) π)}{3 \cdot 3}$

Step 2.3.8.3

Multiply $20$ by $2$ .

$f'' (x) = - \frac{40 π (\cos (\frac{2 π x}{3} + \frac{π}{6}) π)}{3 \cdot 3}$

Step 2.4

Raise $π$ to the power of $1$ .

$f'' (x) = - \frac{40 (π π) \cos (\frac{2 π x}{3} + \frac{π}{6})}{3 \cdot 3}$

Step 2.5

Raise $π$ to the power of $1$ .

$f'' (x) = - \frac{40 (π π) \cos (\frac{2 π x}{3} + \frac{π}{6})}{3 \cdot 3}$

Step 2.6

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

$f'' (x) = - \frac{40 π^{1 + 1} \cos (\frac{2 π x}{3} + \frac{π}{6})}{3 \cdot 3}$

Step 2.7

Add $1$ and $1$ .

$f'' (x) = - \frac{40 π^{2} \cos (\frac{2 π x}{3} + \frac{π}{6})}{3 \cdot 3}$

Step 2.8

Multiply $3$ by $3$ .

$f'' (x) = - \frac{40 π^{2} \cos (\frac{2 π x}{3} + \frac{π}{6})}{9}$

Step 3

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

$- \frac{20 π \sin (\frac{2 π x}{3} + \frac{π}{6})}{3} = 0$

Step 4

Set the numerator equal to zero.

$20 π \sin (\frac{2 π x}{3} + \frac{π}{6}) = 0$

Step 5

Solve the equation for $x$ .

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Step 5.1

Divide each term in $20 π \sin (\frac{2 π x}{3} + \frac{π}{6}) = 0$ by $20 π$ and simplify.

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Step 5.1.1

Divide each term in $20 π \sin (\frac{2 π x}{3} + \frac{π}{6}) = 0$ by $20 π$ .

$\frac{20 π \sin (\frac{2 π x}{3} + \frac{π}{6})}{20 π} = \frac{0}{20 π}$

Step 5.1.2

Simplify the left side.

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Step 5.1.2.1

Cancel the common factor of $20$ .

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Step 5.1.2.1.1

Cancel the common factor.

$\frac{20 π \sin (\frac{2 π x}{3} + \frac{π}{6})}{20 π} = \frac{0}{20 π}$

Step 5.1.2.1.2

Rewrite the expression.

$\frac{π \sin (\frac{2 π x}{3} + \frac{π}{6})}{π} = \frac{0}{20 π}$

Step 5.1.2.2

Cancel the common factor of $π$ .

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Step 5.1.2.2.1

Cancel the common factor.

$\frac{π \sin (\frac{2 π x}{3} + \frac{π}{6})}{π} = \frac{0}{20 π}$

Step 5.1.2.2.2

Divide $\sin (\frac{2 π x}{3} + \frac{π}{6})$ by $1$ .

$\sin (\frac{2 π x}{3} + \frac{π}{6}) = \frac{0}{20 π}$

Step 5.1.3

Simplify the right side.

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Step 5.1.3.1

Cancel the common factor of $0$ and $20$ .

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Step 5.1.3.1.1

Factor $20$ out of $0$ .

$\sin (\frac{2 π x}{3} + \frac{π}{6}) = \frac{20 (0)}{20 π}$

Step 5.1.3.1.2

Cancel the common factors.

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Step 5.1.3.1.2.1

Factor $20$ out of $20 π$ .

$\sin (\frac{2 π x}{3} + \frac{π}{6}) = \frac{20 (0)}{20 (π)}$

Step 5.1.3.1.2.2

Cancel the common factor.

$\sin (\frac{2 π x}{3} + \frac{π}{6}) = \frac{20 \cdot 0}{20 π}$

Step 5.1.3.1.2.3

Rewrite the expression.

$\sin (\frac{2 π x}{3} + \frac{π}{6}) = \frac{0}{π}$

Step 5.1.3.2

Divide $0$ by $π$ .

$\sin (\frac{2 π x}{3} + \frac{π}{6}) = 0$

Step 5.2

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

$\frac{2 π x}{3} + \frac{π}{6} = \arcsin (0)$

Step 5.3

Simplify the right side.

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Step 5.3.1

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

$\frac{2 π x}{3} + \frac{π}{6} = 0$

Step 5.4

Subtract $\frac{π}{6}$ from both sides of the equation.

$\frac{2 π x}{3} = - \frac{π}{6}$

Step 5.5

Multiply both sides of the equation by $\frac{3}{2 π}$ .

$\frac{3}{2 π} \cdot \frac{2 π x}{3} = \frac{3}{2 π} (- \frac{π}{6})$

Step 5.6

Simplify both sides of the equation.

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Step 5.6.1

Simplify the left side.

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Step 5.6.1.1

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

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Step 5.6.1.1.1

Cancel the common factor of $3$ .

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Step 5.6.1.1.1.1

Cancel the common factor.

$\frac{3}{2 π} \cdot \frac{2 π x}{3} = \frac{3}{2 π} (- \frac{π}{6})$

Step 5.6.1.1.1.2

Rewrite the expression.

$\frac{1}{2 π} (2 π x) = \frac{3}{2 π} (- \frac{π}{6})$

Step 5.6.1.1.2

Cancel the common factor of $2 π$ .

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Step 5.6.1.1.2.1

Factor $2 π$ out of $2 π x$ .

$\frac{1}{2 π} (2 π (x)) = \frac{3}{2 π} (- \frac{π}{6})$

Step 5.6.1.1.2.2

Cancel the common factor.

$\frac{1}{2 π} (2 π x) = \frac{3}{2 π} (- \frac{π}{6})$

Step 5.6.1.1.2.3

Rewrite the expression.

$x = \frac{3}{2 π} (- \frac{π}{6})$

Step 5.6.2

Simplify the right side.

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Step 5.6.2.1

Simplify $\frac{3}{2 π} (- \frac{π}{6})$ .

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Step 5.6.2.1.1

Cancel the common factor of $3$ .

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Step 5.6.2.1.1.1

Move the leading negative in $- \frac{π}{6}$ into the numerator.

$x = \frac{3}{2 π} \cdot \frac{- π}{6}$

Step 5.6.2.1.1.2

Factor $3$ out of $6$ .

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

Step 5.6.2.1.1.3

Cancel the common factor.

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

Step 5.6.2.1.1.4

Rewrite the expression.

$x = \frac{1}{2 π} \cdot \frac{- π}{2}$

Step 5.6.2.1.2

Cancel the common factor of $π$ .

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Step 5.6.2.1.2.1

Factor $π$ out of $2 π$ .

$x = \frac{1}{π \cdot 2} \cdot \frac{- π}{2}$

Step 5.6.2.1.2.2

Factor $π$ out of $- π$ .

$x = \frac{1}{π \cdot 2} \cdot \frac{π \cdot - 1}{2}$

Step 5.6.2.1.2.3

Cancel the common factor.

$x = \frac{1}{π \cdot 2} \cdot \frac{π \cdot - 1}{2}$

Step 5.6.2.1.2.4

Rewrite the expression.

$x = \frac{1}{2} \cdot \frac{- 1}{2}$

Step 5.6.2.1.3

Multiply $\frac{1}{2}$ by $\frac{- 1}{2}$ .

$x = \frac{- 1}{2 \cdot 2}$

Step 5.6.2.1.4

Simplify the expression.

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Step 5.6.2.1.4.1

Multiply $2$ by $2$ .

$x = \frac{- 1}{4}$

Step 5.6.2.1.4.2

Move the negative in front of the fraction.

$x = - \frac{1}{4}$

Step 5.7

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.

$\frac{2 π x}{3} + \frac{π}{6} = π - 0$

Step 5.8

Solve for $x$ .

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Step 5.8.1

Subtract $0$ from $π$ .

$\frac{2 π x}{3} + \frac{π}{6} = π$

Step 5.8.2

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

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Step 5.8.2.1

Subtract $\frac{π}{6}$ from both sides of the equation.

$\frac{2 π x}{3} = π - \frac{π}{6}$

Step 5.8.2.2

To write $π$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{2 π x}{3} = π \cdot \frac{6}{6} - \frac{π}{6}$

Step 5.8.2.3

Combine $π$ and $\frac{6}{6}$ .

$\frac{2 π x}{3} = \frac{π \cdot 6}{6} - \frac{π}{6}$

Step 5.8.2.4

Combine the numerators over the common denominator.

$\frac{2 π x}{3} = \frac{π \cdot 6 - π}{6}$

Step 5.8.2.5

Simplify the numerator.

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Step 5.8.2.5.1

Move $6$ to the left of $π$ .

$\frac{2 π x}{3} = \frac{6 \cdot π - π}{6}$

Step 5.8.2.5.2

Subtract $π$ from $6 π$ .

$\frac{2 π x}{3} = \frac{5 π}{6}$

Step 5.8.3

Multiply both sides of the equation by $\frac{3}{2 π}$ .

$\frac{3}{2 π} \cdot \frac{2 π x}{3} = \frac{3}{2 π} \cdot \frac{5 π}{6}$

Step 5.8.4

Simplify both sides of the equation.

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Step 5.8.4.1

Simplify the left side.

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Step 5.8.4.1.1

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

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Step 5.8.4.1.1.1

Cancel the common factor of $3$ .

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Step 5.8.4.1.1.1.1

Cancel the common factor.

$\frac{3}{2 π} \cdot \frac{2 π x}{3} = \frac{3}{2 π} \cdot \frac{5 π}{6}$

Step 5.8.4.1.1.1.2

Rewrite the expression.

$\frac{1}{2 π} (2 π x) = \frac{3}{2 π} \cdot \frac{5 π}{6}$

Step 5.8.4.1.1.2

Cancel the common factor of $2 π$ .

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Step 5.8.4.1.1.2.1

Factor $2 π$ out of $2 π x$ .

$\frac{1}{2 π} (2 π (x)) = \frac{3}{2 π} \cdot \frac{5 π}{6}$

Step 5.8.4.1.1.2.2

Cancel the common factor.

$\frac{1}{2 π} (2 π x) = \frac{3}{2 π} \cdot \frac{5 π}{6}$

Step 5.8.4.1.1.2.3

Rewrite the expression.

$x = \frac{3}{2 π} \cdot \frac{5 π}{6}$

Step 5.8.4.2

Simplify the right side.

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Step 5.8.4.2.1

Simplify $\frac{3}{2 π} \cdot \frac{5 π}{6}$ .

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Step 5.8.4.2.1.1

Cancel the common factor of $3$ .

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Step 5.8.4.2.1.1.1

Factor $3$ out of $6$ .

$x = \frac{3}{2 π} \cdot \frac{5 π}{3 (2)}$

Step 5.8.4.2.1.1.2

Cancel the common factor.

$x = \frac{3}{2 π} \cdot \frac{5 π}{3 \cdot 2}$

Step 5.8.4.2.1.1.3

Rewrite the expression.

$x = \frac{1}{2 π} \cdot \frac{5 π}{2}$

Step 5.8.4.2.1.2

Cancel the common factor of $π$ .

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Step 5.8.4.2.1.2.1

Factor $π$ out of $2 π$ .

$x = \frac{1}{π \cdot 2} \cdot \frac{5 π}{2}$

Step 5.8.4.2.1.2.2

Factor $π$ out of $5 π$ .

$x = \frac{1}{π \cdot 2} \cdot \frac{π \cdot 5}{2}$

Step 5.8.4.2.1.2.3

Cancel the common factor.

$x = \frac{1}{π \cdot 2} \cdot \frac{π \cdot 5}{2}$

Step 5.8.4.2.1.2.4

Rewrite the expression.

$x = \frac{1}{2} \cdot \frac{5}{2}$

Step 5.8.4.2.1.3

Multiply $\frac{1}{2}$ by $\frac{5}{2}$ .

$x = \frac{5}{2 \cdot 2}$

Step 5.8.4.2.1.4

Multiply $2$ by $2$ .

$x = \frac{5}{4}$

Step 5.9

The solution to the equation $\frac{2 π x}{3} + \frac{π}{6} = 0$ .

$x = - \frac{1}{4}, \frac{5}{4}$

Step 6

Evaluate the second derivative at $x = - \frac{1}{4}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- \frac{40 π^{2} \cos (\frac{2 π (- \frac{1}{4})}{3} + \frac{π}{6})}{9}$

Step 7

Evaluate the second derivative.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Multiply $- 1$ by $2$ .

$- \frac{40 π^{2} \cos (\frac{- 2 π \frac{1}{4}}{3} + \frac{π}{6})}{9}$

Step 7.1.2

Combine $\frac{1}{4}$ and $- 2$ .

$- \frac{40 π^{2} \cos (\frac{\frac{- 2}{4} π}{3} + \frac{π}{6})}{9}$

Step 7.1.3

Combine $\frac{- 2}{4}$ and $π$ .

$- \frac{40 π^{2} \cos (\frac{\frac{- 2 π}{4}}{3} + \frac{π}{6})}{9}$

Step 7.2

Simplify the numerator.

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Step 7.2.1

Reduce the expression $\frac{- 2 π}{4}$ by cancelling the common factors.

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Step 7.2.1.1

Factor $2$ out of $- 2 π$ .

$- \frac{40 π^{2} \cos (\frac{\frac{2 (- π)}{4}}{3} + \frac{π}{6})}{9}$

Step 7.2.1.2

Factor $2$ out of $4$ .

$- \frac{40 π^{2} \cos (\frac{\frac{2 (- π)}{2 (2)}}{3} + \frac{π}{6})}{9}$

Step 7.2.1.3

Cancel the common factor.

$- \frac{40 π^{2} \cos (\frac{\frac{2 (- π)}{2 \cdot 2}}{3} + \frac{π}{6})}{9}$

Step 7.2.1.4

Rewrite the expression.

$- \frac{40 π^{2} \cos (\frac{\frac{- π}{2}}{3} + \frac{π}{6})}{9}$

Step 7.2.2

Move the negative in front of the fraction.

$- \frac{40 π^{2} \cos (\frac{- \frac{π}{2}}{3} + \frac{π}{6})}{9}$

Step 7.3

Simplify the numerator.

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Step 7.3.1

Simplify each term.

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Step 7.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$- \frac{40 π^{2} \cos (- \frac{π}{2} \cdot \frac{1}{3} + \frac{π}{6})}{9}$

Step 7.3.1.2

Multiply $- \frac{π}{2} \cdot \frac{1}{3}$ .

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Step 7.3.1.2.1

Multiply $\frac{1}{3}$ by $\frac{π}{2}$ .

$- \frac{40 π^{2} \cos (- \frac{π}{3 \cdot 2} + \frac{π}{6})}{9}$

Step 7.3.1.2.2

Multiply $3$ by $2$ .

$- \frac{40 π^{2} \cos (- \frac{π}{6} + \frac{π}{6})}{9}$

Step 7.3.2

Combine the numerators over the common denominator.

$- \frac{40 π^{2} \cos (\frac{- π + π}{6})}{9}$

Step 7.3.3

Add $- π$ and $π$ .

$- \frac{40 π^{2} \cos (\frac{0}{6})}{9}$

Step 7.3.4

Divide $0$ by $6$ .

$- \frac{40 π^{2} \cos (0)}{9}$

Step 7.3.5

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

$- \frac{40 π^{2} \cdot 1}{9}$

Step 7.3.6

Multiply $40$ by $1$ .

$- \frac{40 π^{2}}{9}$

Step 8

$x = - \frac{1}{4}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = - \frac{1}{4}$ is a local maximum

Step 9

Find the y-value when $x = - \frac{1}{4}$ .

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Step 9.1

Replace the variable $x$ with $- \frac{1}{4}$ in the expression.

$f (- \frac{1}{4}) = 10 \cos (\frac{2 π}{3} \cdot ((- \frac{1}{4}) + \frac{1}{4}))$

Step 9.2

Simplify the result.

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Step 9.2.1

Combine the numerators over the common denominator.

$f (- \frac{1}{4}) = 10 \cos (\frac{2 π}{3} \cdot \frac{- 1 + 1}{4})$

Step 9.2.2

Add $- 1$ and $1$ .

$f (- \frac{1}{4}) = 10 \cos (\frac{2 π}{3} \cdot \frac{0}{4})$

Step 9.2.3

Cancel the common factor of $2$ .

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Step 9.2.3.1

Factor $2$ out of $2 π$ .

$f (- \frac{1}{4}) = 10 \cos (\frac{2 (π)}{3} \cdot \frac{0}{4})$

Step 9.2.3.2

Factor $2$ out of $4$ .

$f (- \frac{1}{4}) = 10 \cos (\frac{2 (π)}{3} \cdot \frac{0}{2 (2)})$

Step 9.2.3.3

Cancel the common factor.

$f (- \frac{1}{4}) = 10 \cos (\frac{2 π}{3} \cdot \frac{0}{2 \cdot 2})$

Step 9.2.3.4

Rewrite the expression.

$f (- \frac{1}{4}) = 10 \cos (\frac{π}{3} \cdot \frac{0}{2})$

Step 9.2.4

Multiply $\frac{π}{3}$ by $\frac{0}{2}$ .

$f (- \frac{1}{4}) = 10 \cos (\frac{π \cdot 0}{3 \cdot 2})$

Step 9.2.5

Simplify the expression.

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Step 9.2.5.1

Multiply $π$ by $0$ .

$f (- \frac{1}{4}) = 10 \cos (\frac{0}{3 \cdot 2})$

Step 9.2.5.2

Multiply $3$ by $2$ .

$f (- \frac{1}{4}) = 10 \cos (\frac{0}{6})$

Step 9.2.5.3

Divide $0$ by $6$ .

$f (- \frac{1}{4}) = 10 \cos (0)$

Step 9.2.6

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

$f (- \frac{1}{4}) = 10 \cdot 1$

Step 9.2.7

Multiply $10$ by $1$ .

$f (- \frac{1}{4}) = 10$

Step 9.2.8

The final answer is $10$ .

$y = 10$

Step 10

Evaluate the second derivative at $x = \frac{5}{4}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- \frac{40 π^{2} \cos (\frac{2 π (\frac{5}{4})}{3} + \frac{π}{6})}{9}$

Step 11

Evaluate the second derivative.

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Step 11.1

Simplify the numerator.

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Step 11.1.1

Combine $\frac{5}{4}$ and $2$ .

$- \frac{40 π^{2} \cos (\frac{\frac{5 \cdot 2}{4} π}{3} + \frac{π}{6})}{9}$

Step 11.1.2

Combine $\frac{5 \cdot 2}{4}$ and $π$ .

$- \frac{40 π^{2} \cos (\frac{\frac{5 \cdot 2 π}{4}}{3} + \frac{π}{6})}{9}$

Step 11.2

Reduce the expression by cancelling the common factors.

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Step 11.2.1

Multiply $5$ by $2$ .

$- \frac{40 π^{2} \cos (\frac{\frac{10 π}{4}}{3} + \frac{π}{6})}{9}$

Step 11.2.2

Reduce the expression $\frac{10 π}{4}$ by cancelling the common factors.

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Step 11.2.2.1

Factor $2$ out of $10 π$ .

$- \frac{40 π^{2} \cos (\frac{\frac{2 (5 π)}{4}}{3} + \frac{π}{6})}{9}$

Step 11.2.2.2

Factor $2$ out of $4$ .

$- \frac{40 π^{2} \cos (\frac{\frac{2 (5 π)}{2 (2)}}{3} + \frac{π}{6})}{9}$

Step 11.2.2.3

Cancel the common factor.

$- \frac{40 π^{2} \cos (\frac{\frac{2 (5 π)}{2 \cdot 2}}{3} + \frac{π}{6})}{9}$

Step 11.2.2.4

Rewrite the expression.

$- \frac{40 π^{2} \cos (\frac{\frac{5 π}{2}}{3} + \frac{π}{6})}{9}$

Step 11.3

Simplify the numerator.

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Step 11.3.1

Simplify each term.

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Step 11.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$- \frac{40 π^{2} \cos (\frac{5 π}{2} \cdot \frac{1}{3} + \frac{π}{6})}{9}$

Step 11.3.1.2

Multiply $\frac{5 π}{2} \cdot \frac{1}{3}$ .

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Step 11.3.1.2.1

Multiply $\frac{5 π}{2}$ by $\frac{1}{3}$ .

$- \frac{40 π^{2} \cos (\frac{5 π}{2 \cdot 3} + \frac{π}{6})}{9}$

Step 11.3.1.2.2

Multiply $2$ by $3$ .

$- \frac{40 π^{2} \cos (\frac{5 π}{6} + \frac{π}{6})}{9}$

Step 11.3.2

Combine the numerators over the common denominator.

$- \frac{40 π^{2} \cos (\frac{5 π + π}{6})}{9}$

Step 11.3.3

Add $5 π$ and $π$ .

$- \frac{40 π^{2} \cos (\frac{6 π}{6})}{9}$

Step 11.3.4

Cancel the common factor of $6$ .

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Step 11.3.4.1

Cancel the common factor.

$- \frac{40 π^{2} \cos (\frac{6 π}{6})}{9}$

Step 11.3.4.2

Divide $π$ by $1$ .

$- \frac{40 π^{2} \cos (π)}{9}$

Step 11.3.5

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.

$- \frac{40 π^{2} (- \cos (0))}{9}$

Step 11.3.6

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

$- \frac{40 π^{2} (- 1 \cdot 1)}{9}$

Step 11.3.7

Multiply $- 1$ by $1$ .

$- \frac{40 π^{2} \cdot - 1}{9}$

Step 11.3.8

Multiply $- 1$ by $40$ .

$- \frac{- 40 π^{2}}{9}$

Step 11.4

Move the negative in front of the fraction.

$- - \frac{40 π^{2}}{9}$

Step 11.5

Multiply $- - \frac{40 π^{2}}{9}$ .

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Step 11.5.1

Multiply $- 1$ by $- 1$ .

$1 \frac{40 π^{2}}{9}$

Step 11.5.2

Multiply $\frac{40 π^{2}}{9}$ by $1$ .

$\frac{40 π^{2}}{9}$

Step 12

$x = \frac{5}{4}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = \frac{5}{4}$ is a local minimum

Step 13

Find the y-value when $x = \frac{5}{4}$ .

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Step 13.1

Replace the variable $x$ with $\frac{5}{4}$ in the expression.

$f (\frac{5}{4}) = 10 \cos (\frac{2 π}{3} \cdot ((\frac{5}{4}) + \frac{1}{4}))$

Step 13.2

Simplify the result.

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Step 13.2.1

Combine the numerators over the common denominator.

$f (\frac{5}{4}) = 10 \cos (\frac{2 π}{3} \cdot \frac{5 + 1}{4})$

Step 13.2.2

Add $5$ and $1$ .

$f (\frac{5}{4}) = 10 \cos (\frac{2 π}{3} \cdot \frac{6}{4})$

Step 13.2.3

Cancel the common factor of $2$ .

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Step 13.2.3.1

Factor $2$ out of $2 π$ .

$f (\frac{5}{4}) = 10 \cos (\frac{2 (π)}{3} \cdot \frac{6}{4})$

Step 13.2.3.2

Factor $2$ out of $4$ .

$f (\frac{5}{4}) = 10 \cos (\frac{2 (π)}{3} \cdot \frac{6}{2 (2)})$

Step 13.2.3.3

Cancel the common factor.

$f (\frac{5}{4}) = 10 \cos (\frac{2 π}{3} \cdot \frac{6}{2 \cdot 2})$

Step 13.2.3.4

Rewrite the expression.

$f (\frac{5}{4}) = 10 \cos (\frac{π}{3} \cdot \frac{6}{2})$

Step 13.2.4

Cancel the common factor of $3$ .

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Step 13.2.4.1

Factor $3$ out of $6$ .

$f (\frac{5}{4}) = 10 \cos (\frac{π}{3} \cdot \frac{3 (2)}{2})$

Step 13.2.4.2

Cancel the common factor.

$f (\frac{5}{4}) = 10 \cos (\frac{π}{3} \cdot \frac{3 \cdot 2}{2})$

Step 13.2.4.3

Rewrite the expression.

$f (\frac{5}{4}) = 10 \cos (π \cdot \frac{2}{2})$

Step 13.2.5

Combine $π$ and $\frac{2}{2}$ .

$f (\frac{5}{4}) = 10 \cos (\frac{π \cdot 2}{2})$

Step 13.2.6

Cancel the common factor of $2$ .

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Step 13.2.6.1

Cancel the common factor.

$f (\frac{5}{4}) = 10 \cos (\frac{π \cdot 2}{2})$

Step 13.2.6.2

Divide $π$ by $1$ .

$f (\frac{5}{4}) = 10 \cos (π)$

Step 13.2.7

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.

$f (\frac{5}{4}) = 10 (- \cos (0))$

Step 13.2.8

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

$f (\frac{5}{4}) = 10 (- 1 \cdot 1)$

Step 13.2.9

Multiply $10 (- 1 \cdot 1)$ .

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Step 13.2.9.1

Multiply $- 1$ by $1$ .

$f (\frac{5}{4}) = 10 \cdot - 1$

Step 13.2.9.2

Multiply $10$ by $- 1$ .

$f (\frac{5}{4}) = - 10$

Step 13.2.10

The final answer is $- 10$ .

$y = - 10$

Step 14

These are the local extrema for $f (x) = 10 \cos (\frac{2 π}{3} \cdot (x + \frac{1}{4}))$ .

$(- \frac{1}{4}, 10)$ is a local maxima

$(\frac{5}{4}, - 10)$ is a local minima

Step 15