Algebra Examples

$y = - 3 + 4 \cos (\frac{5 π}{6} (x + 4))$

Step 1

Find the first derivative of the function.

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Step 1.1

Differentiate.

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Step 1.1.1

By the Sum Rule, the derivative of $- 3 + 4 \cos (\frac{5 π}{6} \cdot (x + 4))$ with respect to $x$ is $\frac{d}{d x} [- 3] + \frac{d}{d x} [4 \cos (\frac{5 π}{6} \cdot (x + 4))]$ .

$\frac{d}{d x} [- 3] + \frac{d}{d x} [4 \cos (\frac{5 π}{6} \cdot (x + 4))]$

Step 1.1.2

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [4 \cos (\frac{5 π}{6} \cdot (x + 4))]$

Step 1.2

Evaluate $\frac{d}{d x} [4 \cos (\frac{5 π}{6} \cdot (x + 4))]$ .

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Step 1.2.1

Since $4$ is constant with respect to $x$ , the derivative of $4 \cos (\frac{5 π}{6} (x + 4))$ with respect to $x$ is $4 \frac{d}{d x} [\cos (\frac{5 π}{6} (x + 4))]$ .

$0 + 4 \frac{d}{d x} [\cos (\frac{5 π}{6} (x + 4))]$

Step 1.2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos (x)$ and $g (x) = \frac{5 π}{6} (x + 4)$ .

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Step 1.2.2.1

To apply the Chain Rule, set $u$ as $\frac{5 π}{6} (x + 4)$ .

$0 + 4 (\frac{d}{d u} [\cos (u)] \frac{d}{d x} [\frac{5 π}{6} (x + 4)])$

Step 1.2.2.2

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

$0 + 4 (- \sin (\frac{5 π}{6} (x + 4)) \frac{d}{d x} [\frac{5 π}{6} (x + 4)])$

Step 1.2.3

Since $\frac{5 π}{6}$ is constant with respect to $x$ , the derivative of $\frac{5 π}{6} (x + 4)$ with respect to $x$ is $\frac{5 π}{6} \frac{d}{d x} [x + 4]$ .

$0 + 4 (- \sin (\frac{5 π}{6} (x + 4)) (\frac{5 π}{6} \frac{d}{d x} [x + 4]))$

Step 1.2.4

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

$0 + 4 (- \sin (\frac{5 π}{6} (x + 4)) (\frac{5 π}{6} (\frac{d}{d x} [x] + \frac{d}{d x} [4])))$

Step 1.2.5

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

$0 + 4 (- \sin (\frac{5 π}{6} (x + 4)) (\frac{5 π}{6} (1 + \frac{d}{d x} [4])))$

Step 1.2.6

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$0 + 4 (- \sin (\frac{5 π}{6} (x + 4)) (\frac{5 π}{6} (1 + 0)))$

Step 1.2.7

Add $1$ and $0$ .

$0 + 4 (- \sin (\frac{5 π}{6} (x + 4)) (\frac{5 π}{6} \cdot 1))$

Step 1.2.8

Multiply $\frac{5 π}{6}$ by $1$ .

$0 + 4 (- \sin (\frac{5 π}{6} (x + 4)) \frac{5 π}{6})$

Step 1.2.9

Combine $\frac{5 π}{6}$ and $\sin (\frac{5 π}{6} (x + 4))$ .

$0 + 4 (- \frac{5 π \sin (\frac{5 π}{6} (x + 4))}{6})$

Step 1.2.10

Multiply $- 1$ by $4$ .

$0 - 4 \frac{5 π \sin (\frac{5 π}{6} (x + 4))}{6}$

Step 1.2.11

Combine $- 4$ and $\frac{5 π \sin (\frac{5 π}{6} (x + 4))}{6}$ .

$0 + \frac{- 4 (5 π \sin (\frac{5 π}{6} (x + 4)))}{6}$

Step 1.2.12

Multiply $5$ by $- 4$ .

$0 + \frac{- 20 (π \sin (\frac{5 π}{6} (x + 4)))}{6}$

Step 1.2.13

Cancel the common factor of $- 20$ and $6$ .

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Step 1.2.13.1

Factor $2$ out of $- 20 π \sin (\frac{5 π}{6} (x + 4))$ .

$0 + \frac{2 (- 10 π \sin (\frac{5 π}{6} (x + 4)))}{6}$

Step 1.2.13.2

Cancel the common factors.

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Step 1.2.13.2.1

Factor $2$ out of $6$ .

$0 + \frac{2 (- 10 π \sin (\frac{5 π}{6} (x + 4)))}{2 (3)}$

Step 1.2.13.2.2

Cancel the common factor.

$0 + \frac{2 (- 10 π \sin (\frac{5 π}{6} (x + 4)))}{2 \cdot 3}$

Step 1.2.13.2.3

Rewrite the expression.

$0 + \frac{- 10 π \sin (\frac{5 π}{6} (x + 4))}{3}$

Step 1.2.14

Move the negative in front of the fraction.

$0 - \frac{10 π \sin (\frac{5 π}{6} (x + 4))}{3}$

Step 1.3

Simplify.

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Step 1.3.1

Apply the distributive property.

$0 - \frac{10 π \sin (\frac{5 π}{6} x + \frac{5 π}{6} \cdot 4)}{3}$

Step 1.3.2

Combine terms.

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Step 1.3.2.1

Combine $\frac{5 π}{6}$ and $4$ .

$0 - \frac{10 π \sin (\frac{5 π}{6} x + \frac{5 π \cdot 4}{6})}{3}$

Step 1.3.2.2

Multiply $4$ by $5$ .

$0 - \frac{10 π \sin (\frac{5 π}{6} x + \frac{20 π}{6})}{3}$

Step 1.3.2.3

Cancel the common factor of $20$ and $6$ .

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Step 1.3.2.3.1

Factor $2$ out of $20 π$ .

$0 - \frac{10 π \sin (\frac{5 π}{6} x + \frac{2 (10 π)}{6})}{3}$

Step 1.3.2.3.2

Cancel the common factors.

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Step 1.3.2.3.2.1

Factor $2$ out of $6$ .

$0 - \frac{10 π \sin (\frac{5 π}{6} x + \frac{2 (10 π)}{2 (3)})}{3}$

Step 1.3.2.3.2.2

Cancel the common factor.

$0 - \frac{10 π \sin (\frac{5 π}{6} x + \frac{2 (10 π)}{2 \cdot 3})}{3}$

Step 1.3.2.3.2.3

Rewrite the expression.

$0 - \frac{10 π \sin (\frac{5 π}{6} x + \frac{10 π}{3})}{3}$

Step 1.3.2.4

Combine $\frac{5 π}{6}$ and $x$ .

$0 - \frac{10 π \sin (\frac{5 π x}{6} + \frac{10 π}{3})}{3}$

Step 1.3.2.5

Subtract $\frac{10 π \sin (\frac{5 π x}{6} + \frac{10 π}{3})}{3}$ from $0$ .

$- \frac{10 π \sin (\frac{5 π x}{6} + \frac{10 π}{3})}{3}$

Step 2

Find the second derivative of the function.

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Step 2.1

Since $- \frac{10 π}{3}$ is constant with respect to $x$ , the derivative of $- \frac{10 π \sin (\frac{5 π x}{6} + \frac{10 π}{3})}{3}$ with respect to $x$ is $- \frac{10 π}{3} \frac{d}{d x} [\sin (\frac{5 π x}{6} + \frac{10 π}{3})]$ .

$f'' (x) = - \frac{10 π}{3} \cdot \frac{d}{d x} \sin (\frac{5 π x}{6} + \frac{10 π}{3})$

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 $g (x) = \frac{5 π x}{6} + \frac{10 π}{3}$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $\frac{5 π x}{6} + \frac{10 π}{3}$ .

$f'' (x) = - \frac{10 π}{3} \cdot (\frac{d}{d u} (\sin (u)) \frac{d}{d x} (\frac{5 π x}{6} + \frac{10 π}{3}))$

Step 2.2.2

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

$f'' (x) = - \frac{10 π}{3} \cdot (\cos (u) \frac{d}{d x} (\frac{5 π x}{6} + \frac{10 π}{3}))$

Step 2.2.3

Replace all occurrences of $u$ with $\frac{5 π x}{6} + \frac{10 π}{3}$ .

$f'' (x) = - \frac{10 π}{3} \cdot (\cos (\frac{5 π x}{6} + \frac{10 π}{3}) \frac{d}{d x} (\frac{5 π x}{6} + \frac{10 π}{3}))$

Step 2.3

Differentiate.

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Step 2.3.1

Combine $\cos (\frac{5 π x}{6} + \frac{10 π}{3})$ and $\frac{10 π}{3}$ .

$f'' (x) = - \frac{\cos (\frac{5 π x}{6} + \frac{10 π}{3}) (10 π)}{3} \cdot \frac{d}{d x} (\frac{5 π x}{6} + \frac{10 π}{3})$

Step 2.3.2

Move $10$ to the left of $\cos (\frac{5 π x}{6} + \frac{10 π}{3})$ .

$f'' (x) = - \frac{10 \cdot (\cos (\frac{5 π x}{6} + \frac{10 π}{3}) π)}{3} \cdot \frac{d}{d x} (\frac{5 π x}{6} + \frac{10 π}{3})$

Step 2.3.3

By the Sum Rule, the derivative of $\frac{5 π x}{6} + \frac{10 π}{3}$ with respect to $x$ is $\frac{d}{d x} [\frac{5 π x}{6}] + \frac{d}{d x} [\frac{10 π}{3}]$ .

$f'' (x) = - \frac{10 \cos (\frac{5 π x}{6} + \frac{10 π}{3}) π}{3} \cdot (\frac{d}{d x} (\frac{5 π x}{6}) + \frac{d}{d x} (\frac{10 π}{3}))$

Step 2.3.4

Since $\frac{5 π}{6}$ is constant with respect to $x$ , the derivative of $\frac{5 π x}{6}$ with respect to $x$ is $\frac{5 π}{6} \frac{d}{d x} [x]$ .

$f'' (x) = - \frac{10 \cos (\frac{5 π x}{6} + \frac{10 π}{3}) π}{3} \cdot (\frac{5 π}{6} \cdot \frac{d}{d x} (x) + \frac{d}{d x} (\frac{10 π}{3}))$

Step 2.3.5

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

$f'' (x) = - \frac{10 \cos (\frac{5 π x}{6} + \frac{10 π}{3}) π}{3} \cdot (\frac{5 π}{6} \cdot 1 + \frac{d}{d x} (\frac{10 π}{3}))$

Step 2.3.6

Multiply $\frac{5 π}{6}$ by $1$ .

$f'' (x) = - \frac{10 \cos (\frac{5 π x}{6} + \frac{10 π}{3}) π}{3} \cdot (\frac{5 π}{6} + \frac{d}{d x} (\frac{10 π}{3}))$

Step 2.3.7

Since $\frac{10 π}{3}$ is constant with respect to $x$ , the derivative of $\frac{10 π}{3}$ with respect to $x$ is $0$ .

$f'' (x) = - \frac{10 \cos (\frac{5 π x}{6} + \frac{10 π}{3}) π}{3} \cdot (\frac{5 π}{6} + 0)$

Step 2.3.8

Combine fractions.

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Step 2.3.8.1

Add $\frac{5 π}{6}$ and $0$ .

$f'' (x) = - \frac{10 \cos (\frac{5 π x}{6} + \frac{10 π}{3}) π}{3} \cdot \frac{5 π}{6}$

Step 2.3.8.2

Multiply $\frac{5 π}{6}$ by $\frac{10 \cos (\frac{5 π x}{6} + \frac{10 π}{3}) π}{3}$ .

$f'' (x) = - \frac{5 π (10 \cos (\frac{5 π x}{6} + \frac{10 π}{3}) π)}{6 \cdot 3}$

Step 2.3.8.3

Multiply $10$ by $5$ .

$f'' (x) = - \frac{50 π (\cos (\frac{5 π x}{6} + \frac{10 π}{3}) π)}{6 \cdot 3}$

Step 2.4

Raise $π$ to the power of $1$ .

$f'' (x) = - \frac{50 (π π) \cos (\frac{5 π x}{6} + \frac{10 π}{3})}{6 \cdot 3}$

Step 2.5

Raise $π$ to the power of $1$ .

$f'' (x) = - \frac{50 (π π) \cos (\frac{5 π x}{6} + \frac{10 π}{3})}{6 \cdot 3}$

Step 2.6

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

$f'' (x) = - \frac{50 π^{1 + 1} \cos (\frac{5 π x}{6} + \frac{10 π}{3})}{6 \cdot 3}$

Step 2.7

Simplify the expression.

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Step 2.7.1

Add $1$ and $1$ .

$f'' (x) = - \frac{50 π^{2} \cos (\frac{5 π x}{6} + \frac{10 π}{3})}{6 \cdot 3}$

Step 2.7.2

Multiply $6$ by $3$ .

$f'' (x) = - \frac{50 π^{2} \cos (\frac{5 π x}{6} + \frac{10 π}{3})}{18}$

Step 2.8

Cancel the common factor of $50$ and $18$ .

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Step 2.8.1

Factor $2$ out of $50 π^{2} \cos (\frac{5 π x}{6} + \frac{10 π}{3})$ .

$f'' (x) = - \frac{2 (25 π^{2} \cos (\frac{5 π x}{6} + \frac{10 π}{3}))}{18}$

Step 2.8.2

Cancel the common factors.

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Step 2.8.2.1

Factor $2$ out of $18$ .

$f'' (x) = - \frac{2 (25 π^{2} \cos (\frac{5 π x}{6} + \frac{10 π}{3}))}{2 (9)}$

Step 2.8.2.2

Cancel the common factor.

$f'' (x) = - \frac{2 (25 π^{2} \cos (\frac{5 π x}{6} + \frac{10 π}{3}))}{2 \cdot 9}$

Step 2.8.2.3

Rewrite the expression.

$f'' (x) = - \frac{25 π^{2} \cos (\frac{5 π x}{6} + \frac{10 π}{3})}{9}$

Step 3

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

$- \frac{10 π \sin (\frac{5 π x}{6} + \frac{10 π}{3})}{3} = 0$

Step 4

Set the numerator equal to zero.

$10 π \sin (\frac{5 π x}{6} + \frac{10 π}{3}) = 0$

Step 5

Solve the equation for $x$ .

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Step 5.1

Divide each term in $10 π \sin (\frac{5 π x}{6} + \frac{10 π}{3}) = 0$ by $10 π$ and simplify.

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Step 5.1.1

Divide each term in $10 π \sin (\frac{5 π x}{6} + \frac{10 π}{3}) = 0$ by $10 π$ .

$\frac{10 π \sin (\frac{5 π x}{6} + \frac{10 π}{3})}{10 π} = \frac{0}{10 π}$

Step 5.1.2

Simplify the left side.

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Step 5.1.2.1

Cancel the common factor of $10$ .

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Step 5.1.2.1.1

Cancel the common factor.

$\frac{10 π \sin (\frac{5 π x}{6} + \frac{10 π}{3})}{10 π} = \frac{0}{10 π}$

Step 5.1.2.1.2

Rewrite the expression.

$\frac{π \sin (\frac{5 π x}{6} + \frac{10 π}{3})}{π} = \frac{0}{10 π}$

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.

$\frac{π \sin (\frac{5 π x}{6} + \frac{10 π}{3})}{π} = \frac{0}{10 π}$

Step 5.1.2.2.2

Divide $\sin (\frac{5 π x}{6} + \frac{10 π}{3})$ by $1$ .

$\sin (\frac{5 π x}{6} + \frac{10 π}{3}) = \frac{0}{10 π}$

Step 5.1.3

Simplify the right side.

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Step 5.1.3.1

Cancel the common factor of $0$ and $10$ .

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Step 5.1.3.1.1

Factor $10$ out of $0$ .

$\sin (\frac{5 π x}{6} + \frac{10 π}{3}) = \frac{10 (0)}{10 π}$

Step 5.1.3.1.2

Cancel the common factors.

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Step 5.1.3.1.2.1

Factor $10$ out of $10 π$ .

$\sin (\frac{5 π x}{6} + \frac{10 π}{3}) = \frac{10 (0)}{10 (π)}$

Step 5.1.3.1.2.2

Cancel the common factor.

$\sin (\frac{5 π x}{6} + \frac{10 π}{3}) = \frac{10 \cdot 0}{10 π}$

Step 5.1.3.1.2.3

Rewrite the expression.

$\sin (\frac{5 π x}{6} + \frac{10 π}{3}) = \frac{0}{π}$

Step 5.1.3.2

Divide $0$ by $π$ .

$\sin (\frac{5 π x}{6} + \frac{10 π}{3}) = 0$

Step 5.2

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

$\frac{5 π x}{6} + \frac{10 π}{3} = \arcsin (0)$

Step 5.3

Simplify the right side.

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Step 5.3.1

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

$\frac{5 π x}{6} + \frac{10 π}{3} = 0$

Step 5.4

Subtract $\frac{10 π}{3}$ from both sides of the equation.

$\frac{5 π x}{6} = - \frac{10 π}{3}$

Step 5.5

Multiply both sides of the equation by $\frac{6}{5 π}$ .

$\frac{6}{5 π} \cdot \frac{5 π x}{6} = \frac{6}{5 π} (- \frac{10 π}{3})$

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 $\frac{6}{5 π} \cdot \frac{5 π x}{6}$ .

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Step 5.6.1.1.1

Cancel the common factor of $6$ .

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Step 5.6.1.1.1.1

Cancel the common factor.

$\frac{6}{5 π} \cdot \frac{5 π x}{6} = \frac{6}{5 π} (- \frac{10 π}{3})$

Step 5.6.1.1.1.2

Rewrite the expression.

$\frac{1}{5 π} (5 π x) = \frac{6}{5 π} (- \frac{10 π}{3})$

Step 5.6.1.1.2

Cancel the common factor of $5 π$ .

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Step 5.6.1.1.2.1

Factor $5 π$ out of $5 π x$ .

$\frac{1}{5 π} (5 π (x)) = \frac{6}{5 π} (- \frac{10 π}{3})$

Step 5.6.1.1.2.2

Cancel the common factor.

$\frac{1}{5 π} (5 π x) = \frac{6}{5 π} (- \frac{10 π}{3})$

Step 5.6.1.1.2.3

Rewrite the expression.

$x = \frac{6}{5 π} (- \frac{10 π}{3})$

Step 5.6.2

Simplify the right side.

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Step 5.6.2.1

Simplify $\frac{6}{5 π} (- \frac{10 π}{3})$ .

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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 $- \frac{10 π}{3}$ into the numerator.

$x = \frac{6}{5 π} \cdot \frac{- 10 π}{3}$

Step 5.6.2.1.1.2

Factor $3$ out of $6$ .

$x = \frac{3 (2)}{5 π} \cdot \frac{- 10 π}{3}$

Step 5.6.2.1.1.3

Cancel the common factor.

$x = \frac{3 \cdot 2}{5 π} \cdot \frac{- 10 π}{3}$

Step 5.6.2.1.1.4

Rewrite the expression.

$x = \frac{2}{5 π} (- 10 π)$

Step 5.6.2.1.2

Cancel the common factor of $5 π$ .

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Step 5.6.2.1.2.1

Factor $5 π$ out of $- 10 π$ .

$x = \frac{2}{5 π} (5 π (- 2))$

Step 5.6.2.1.2.2

Cancel the common factor.

$x = \frac{2}{5 π} (5 π \cdot - 2)$

Step 5.6.2.1.2.3

Rewrite the expression.

$x = 2 \cdot - 2$

Step 5.6.2.1.3

Multiply $2$ by $- 2$ .

$x = - 4$

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.

$\frac{5 π x}{6} + \frac{10 π}{3} = π - 0$

Step 5.8

Solve for $x$ .

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Step 5.8.1

Subtract $0$ from $π$ .

$\frac{5 π x}{6} + \frac{10 π}{3} = π$

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 $\frac{10 π}{3}$ from both sides of the equation.

$\frac{5 π x}{6} = π - \frac{10 π}{3}$

Step 5.8.2.2

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

$\frac{5 π x}{6} = π \cdot \frac{3}{3} - \frac{10 π}{3}$

Step 5.8.2.3

Combine $π$ and $\frac{3}{3}$ .

$\frac{5 π x}{6} = \frac{π \cdot 3}{3} - \frac{10 π}{3}$

Step 5.8.2.4

Combine the numerators over the common denominator.

$\frac{5 π x}{6} = \frac{π \cdot 3 - 10 π}{3}$

Step 5.8.2.5

Simplify the numerator.

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Step 5.8.2.5.1

Move $3$ to the left of $π$ .

$\frac{5 π x}{6} = \frac{3 \cdot π - 10 π}{3}$

Step 5.8.2.5.2

Subtract $10 π$ from $3 π$ .

$\frac{5 π x}{6} = \frac{- 7 π}{3}$

Step 5.8.2.6

Move the negative in front of the fraction.

$\frac{5 π x}{6} = - \frac{7 π}{3}$

Step 5.8.3

Multiply both sides of the equation by $\frac{6}{5 π}$ .

$\frac{6}{5 π} \cdot \frac{5 π x}{6} = \frac{6}{5 π} (- \frac{7 π}{3})$

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 $\frac{6}{5 π} \cdot \frac{5 π x}{6}$ .

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Step 5.8.4.1.1.1

Cancel the common factor of $6$ .

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Step 5.8.4.1.1.1.1

Cancel the common factor.

$\frac{6}{5 π} \cdot \frac{5 π x}{6} = \frac{6}{5 π} (- \frac{7 π}{3})$

Step 5.8.4.1.1.1.2

Rewrite the expression.

$\frac{1}{5 π} (5 π x) = \frac{6}{5 π} (- \frac{7 π}{3})$

Step 5.8.4.1.1.2

Cancel the common factor of $5 π$ .

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Step 5.8.4.1.1.2.1

Factor $5 π$ out of $5 π x$ .

$\frac{1}{5 π} (5 π (x)) = \frac{6}{5 π} (- \frac{7 π}{3})$

Step 5.8.4.1.1.2.2

Cancel the common factor.

$\frac{1}{5 π} (5 π x) = \frac{6}{5 π} (- \frac{7 π}{3})$

Step 5.8.4.1.1.2.3

Rewrite the expression.

$x = \frac{6}{5 π} (- \frac{7 π}{3})$

Step 5.8.4.2

Simplify the right side.

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Step 5.8.4.2.1

Simplify $\frac{6}{5 π} (- \frac{7 π}{3})$ .

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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

Move the leading negative in $- \frac{7 π}{3}$ into the numerator.

$x = \frac{6}{5 π} \cdot \frac{- 7 π}{3}$

Step 5.8.4.2.1.1.2

Factor $3$ out of $6$ .

$x = \frac{3 (2)}{5 π} \cdot \frac{- 7 π}{3}$

Step 5.8.4.2.1.1.3

Cancel the common factor.

$x = \frac{3 \cdot 2}{5 π} \cdot \frac{- 7 π}{3}$

Step 5.8.4.2.1.1.4

Rewrite the expression.

$x = \frac{2}{5 π} (- 7 π)$

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 $5 π$ .

$x = \frac{2}{π \cdot 5} (- 7 π)$

Step 5.8.4.2.1.2.2

Factor $π$ out of $- 7 π$ .

$x = \frac{2}{π \cdot 5} (π \cdot - 7)$

Step 5.8.4.2.1.2.3

Cancel the common factor.

$x = \frac{2}{π \cdot 5} (π \cdot - 7)$

Step 5.8.4.2.1.2.4

Rewrite the expression.

$x = \frac{2}{5} \cdot - 7$

Step 5.8.4.2.1.3

Combine $\frac{2}{5}$ and $- 7$ .

$x = \frac{2 \cdot - 7}{5}$

Step 5.8.4.2.1.4

Simplify the expression.

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Step 5.8.4.2.1.4.1

Multiply $2$ by $- 7$ .

$x = \frac{- 14}{5}$

Step 5.8.4.2.1.4.2

Move the negative in front of the fraction.

$x = - \frac{14}{5}$

Step 5.9

The solution to the equation $\frac{5 π x}{6} + \frac{10 π}{3} = 0$ .

$x = - 4, - \frac{14}{5}$

Step 6

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

$- \frac{25 π^{2} \cos (\frac{5 π (- 4)}{6} + \frac{10 π}{3})}{9}$

Step 7

Evaluate the second derivative.

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Step 7.1

Reduce the expression by cancelling the common factors.

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Step 7.1.1

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

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Step 7.1.1.1

Factor $2$ out of $5 π (- 4)$ .

$- \frac{25 π^{2} \cos (\frac{2 (5 π \cdot (- 2))}{6} + \frac{10 π}{3})}{9}$

Step 7.1.1.2

Cancel the common factors.

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Step 7.1.1.2.1

Factor $2$ out of $6$ .

$- \frac{25 π^{2} \cos (\frac{2 (5 π \cdot (- 2))}{2 (3)} + \frac{10 π}{3})}{9}$

Step 7.1.1.2.2

Cancel the common factor.

$- \frac{25 π^{2} \cos (\frac{2 (5 π \cdot (- 2))}{2 \cdot 3} + \frac{10 π}{3})}{9}$

Step 7.1.1.2.3

Rewrite the expression.

$- \frac{25 π^{2} \cos (\frac{5 π \cdot (- 2)}{3} + \frac{10 π}{3})}{9}$

Step 7.1.2

Multiply $- 2$ by $5$ .

$- \frac{25 π^{2} \cos (\frac{- 10 π}{3} + \frac{10 π}{3})}{9}$

Step 7.2

Simplify the numerator.

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Step 7.2.1

Combine the numerators over the common denominator.

$- \frac{25 π^{2} \cos (\frac{- 10 π + 10 π}{3})}{9}$

Step 7.2.2

Add $- 10 π$ and $10 π$ .

$- \frac{25 π^{2} \cos (\frac{0}{3})}{9}$

Step 7.2.3

Divide $0$ by $3$ .

$- \frac{25 π^{2} \cos (0)}{9}$

Step 7.2.4

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

$- \frac{25 π^{2} \cdot 1}{9}$

Step 7.2.5

Multiply $25$ by $1$ .

$- \frac{25 π^{2}}{9}$

Step 8

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

$x = - 4$ is a local maximum

Step 9

Find the y-value when $x = - 4$ .

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Step 9.1

Replace the variable $x$ with $- 4$ in the expression.

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

Step 9.2

Simplify the result.

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Step 9.2.1

Simplify each term.

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Step 9.2.1.1

Add $- 4$ and $4$ .

$f (- 4) = - 3 + 4 \cos (\frac{5 π}{6} \cdot 0)$

Step 9.2.1.2

Multiply $\frac{5 π}{6}$ by $0$ .

$f (- 4) = - 3 + 4 \cos (0)$

Step 9.2.1.3

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

$f (- 4) = - 3 + 4 \cdot 1$

Step 9.2.1.4

Multiply $4$ by $1$ .

$f (- 4) = - 3 + 4$

Step 9.2.2

Add $- 3$ and $4$ .

$f (- 4) = 1$

Step 9.2.3

The final answer is $1$ .

$y = 1$

Step 10

Evaluate the second derivative at $x = - \frac{14}{5}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- \frac{25 π^{2} \cos (\frac{5 π (- \frac{14}{5})}{6} + \frac{10 π}{3})}{9}$

Step 11

Evaluate the second derivative.

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Step 11.1

Simplify the numerator.

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Step 11.1.1

Multiply $- 1$ by $5$ .

$- \frac{25 π^{2} \cos (\frac{- 5 π \frac{14}{5}}{6} + \frac{10 π}{3})}{9}$

Step 11.1.2

Combine $\frac{14}{5}$ and $- 5$ .

$- \frac{25 π^{2} \cos (\frac{\frac{14 \cdot - 5}{5} π}{6} + \frac{10 π}{3})}{9}$

Step 11.1.3

Combine $\frac{14 \cdot - 5}{5}$ and $π$ .

$- \frac{25 π^{2} \cos (\frac{\frac{14 \cdot - 5 π}{5}}{6} + \frac{10 π}{3})}{9}$

Step 11.2

Multiply $14$ by $- 5$ .

$- \frac{25 π^{2} \cos (\frac{\frac{- 70 π}{5}}{6} + \frac{10 π}{3})}{9}$

Step 11.3

Reduce the expression by cancelling the common factors.

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Step 11.3.1

Reduce the expression $\frac{- 70 π}{5}$ by cancelling the common factors.

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Step 11.3.1.1

Factor $5$ out of $- 70 π$ .

$- \frac{25 π^{2} \cos (\frac{\frac{5 (- 14 π)}{5}}{6} + \frac{10 π}{3})}{9}$

Step 11.3.1.2

Factor $5$ out of $5$ .

$- \frac{25 π^{2} \cos (\frac{\frac{5 (- 14 π)}{5 (1)}}{6} + \frac{10 π}{3})}{9}$

Step 11.3.1.3

Cancel the common factor.

$- \frac{25 π^{2} \cos (\frac{\frac{5 (- 14 π)}{5 \cdot 1}}{6} + \frac{10 π}{3})}{9}$

Step 11.3.1.4

Rewrite the expression.

$- \frac{25 π^{2} \cos (\frac{\frac{- 14 π}{1}}{6} + \frac{10 π}{3})}{9}$

Step 11.3.2

Divide $- 14 π$ by $1$ .

$- \frac{25 π^{2} \cos (\frac{- 14 π}{6} + \frac{10 π}{3})}{9}$

Step 11.4

Simplify the numerator.

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Step 11.4.1

Simplify each term.

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Step 11.4.1.1

Cancel the common factor of $- 14$ and $6$ .

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Step 11.4.1.1.1

Factor $2$ out of $- 14 π$ .

$- \frac{25 π^{2} \cos (\frac{2 (- 7 π)}{6} + \frac{10 π}{3})}{9}$

Step 11.4.1.1.2

Cancel the common factors.

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Step 11.4.1.1.2.1

Factor $2$ out of $6$ .

$- \frac{25 π^{2} \cos (\frac{2 (- 7 π)}{2 (3)} + \frac{10 π}{3})}{9}$

Step 11.4.1.1.2.2

Cancel the common factor.

$- \frac{25 π^{2} \cos (\frac{2 (- 7 π)}{2 \cdot 3} + \frac{10 π}{3})}{9}$

Step 11.4.1.1.2.3

Rewrite the expression.

$- \frac{25 π^{2} \cos (\frac{- 7 π}{3} + \frac{10 π}{3})}{9}$

Step 11.4.1.2

Move the negative in front of the fraction.

$- \frac{25 π^{2} \cos (- \frac{7 π}{3} + \frac{10 π}{3})}{9}$

Step 11.4.2

Combine the numerators over the common denominator.

$- \frac{25 π^{2} \cos (\frac{- 7 π + 10 π}{3})}{9}$

Step 11.4.3

Add $- 7 π$ and $10 π$ .

$- \frac{25 π^{2} \cos (\frac{3 π}{3})}{9}$

Step 11.4.4

Cancel the common factor of $3$ .

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Step 11.4.4.1

Cancel the common factor.

$- \frac{25 π^{2} \cos (\frac{3 π}{3})}{9}$

Step 11.4.4.2

Divide $π$ by $1$ .

$- \frac{25 π^{2} \cos (π)}{9}$

Step 11.4.5

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.

$- \frac{25 π^{2} (- \cos (0))}{9}$

Step 11.4.6

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

$- \frac{25 π^{2} (- 1 \cdot 1)}{9}$

Step 11.4.7

Multiply $- 1$ by $1$ .

$- \frac{25 π^{2} \cdot - 1}{9}$

Step 11.4.8

Multiply $- 1$ by $25$ .

$- \frac{- 25 π^{2}}{9}$

Step 11.5

Move the negative in front of the fraction.

$- - \frac{25 π^{2}}{9}$

Step 11.6

Multiply $- - \frac{25 π^{2}}{9}$ .

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Step 11.6.1

Multiply $- 1$ by $- 1$ .

$1 \frac{25 π^{2}}{9}$

Step 11.6.2

Multiply $\frac{25 π^{2}}{9}$ by $1$ .

$\frac{25 π^{2}}{9}$

Step 12

$x = - \frac{14}{5}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = - \frac{14}{5}$ is a local minimum

Step 13

Find the y-value when $x = - \frac{14}{5}$ .

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Step 13.1

Replace the variable $x$ with $- \frac{14}{5}$ in the expression.

$f (- \frac{14}{5}) = - 3 + 4 \cos (\frac{5 π}{6} \cdot ((- \frac{14}{5}) + 4))$

Step 13.2

Simplify the result.

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Step 13.2.1

Simplify each term.

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Step 13.2.1.1

To write $4$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$f (- \frac{14}{5}) = - 3 + 4 \cos (\frac{5 π}{6} \cdot (- \frac{14}{5} + 4 \cdot \frac{5}{5}))$

Step 13.2.1.2

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

$f (- \frac{14}{5}) = - 3 + 4 \cos (\frac{5 π}{6} \cdot (- \frac{14}{5} + \frac{4 \cdot 5}{5}))$

Step 13.2.1.3

Combine the numerators over the common denominator.

$f (- \frac{14}{5}) = - 3 + 4 \cos (\frac{5 π}{6} \cdot \frac{- 14 + 4 \cdot 5}{5})$

Step 13.2.1.4

Simplify the numerator.

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Step 13.2.1.4.1

Multiply $4$ by $5$ .

$f (- \frac{14}{5}) = - 3 + 4 \cos (\frac{5 π}{6} \cdot \frac{- 14 + 20}{5})$

Step 13.2.1.4.2

Add $- 14$ and $20$ .

$f (- \frac{14}{5}) = - 3 + 4 \cos (\frac{5 π}{6} \cdot \frac{6}{5})$

Step 13.2.1.5

Cancel the common factor of $5$ .

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Step 13.2.1.5.1

Factor $5$ out of $5 π$ .

$f (- \frac{14}{5}) = - 3 + 4 \cos (\frac{5 (π)}{6} \cdot \frac{6}{5})$

Step 13.2.1.5.2

Cancel the common factor.

$f (- \frac{14}{5}) = - 3 + 4 \cos (\frac{5 π}{6} \cdot \frac{6}{5})$

Step 13.2.1.5.3

Rewrite the expression.

$f (- \frac{14}{5}) = - 3 + 4 \cos (\frac{π}{6} \cdot 6)$

Step 13.2.1.6

Cancel the common factor of $6$ .

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Step 13.2.1.6.1

Cancel the common factor.

$f (- \frac{14}{5}) = - 3 + 4 \cos (\frac{π}{6} \cdot 6)$

Step 13.2.1.6.2

Rewrite the expression.

$f (- \frac{14}{5}) = - 3 + 4 \cos (π)$

Step 13.2.1.7

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.

$f (- \frac{14}{5}) = - 3 + 4 (- \cos (0))$

Step 13.2.1.8

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

$f (- \frac{14}{5}) = - 3 + 4 (- 1 \cdot 1)$

Step 13.2.1.9

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

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Step 13.2.1.9.1

Multiply $- 1$ by $1$ .

$f (- \frac{14}{5}) = - 3 + 4 \cdot - 1$

Step 13.2.1.9.2

Multiply $4$ by $- 1$ .

$f (- \frac{14}{5}) = - 3 - 4$

Step 13.2.2

Subtract $4$ from $- 3$ .

$f (- \frac{14}{5}) = - 7$

Step 13.2.3

The final answer is $- 7$ .

$y = - 7$

Step 14

These are the local extrema for $f (x) = - 3 + 4 \cos (\frac{5 π}{6} \cdot (x + 4))$ .

$(- 4, 1)$ is a local maxima

$(- \frac{14}{5}, - 7)$ is a local minima

Step 15