Calculus Examples

$y = - 2 + 5 \sin (\frac{π}{12} \cdot (x - 2))$

Step 1

Write $y = - 2 + 5 \sin (\frac{π}{12} \cdot (x - 2))$ as a function.

$f (x) = - 2 + 5 \sin (\frac{π}{12} \cdot (x - 2))$

Step 2

Find the first derivative of the function.

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Step 2.1

Differentiate.

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Step 2.1.1

By the Sum Rule, the derivative of $- 2 + 5 \sin (\frac{π}{12} \cdot (x - 2))$ with respect to $x$ is $\frac{d}{d x} [- 2] + \frac{d}{d x} [5 \sin (\frac{π}{12} \cdot (x - 2))]$ .

$\frac{d}{d x} [- 2] + \frac{d}{d x} [5 \sin (\frac{π}{12} \cdot (x - 2))]$

Step 2.1.2

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [5 \sin (\frac{π}{12} \cdot (x - 2))]$

Step 2.2

Evaluate $\frac{d}{d x} [5 \sin (\frac{π}{12} \cdot (x - 2))]$ .

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Step 2.2.1

Since $5$ is constant with respect to $x$ , the derivative of $5 \sin (\frac{π}{12} (x - 2))$ with respect to $x$ is $5 \frac{d}{d x} [\sin (\frac{π}{12} (x - 2))]$ .

$0 + 5 \frac{d}{d x} [\sin (\frac{π}{12} (x - 2))]$

Step 2.2.2

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 $g (x) = \frac{π}{12} (x - 2)$ .

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Step 2.2.2.1

To apply the Chain Rule, set $u$ as $\frac{π}{12} (x - 2)$ .

$0 + 5 (\frac{d}{d u} [\sin (u)] \frac{d}{d x} [\frac{π}{12} (x - 2)])$

Step 2.2.2.2

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

$0 + 5 (\cos (\frac{π}{12} (x - 2)) \frac{d}{d x} [\frac{π}{12} (x - 2)])$

Step 2.2.3

Since $\frac{π}{12}$ is constant with respect to $x$ , the derivative of $\frac{π}{12} (x - 2)$ with respect to $x$ is $\frac{π}{12} \frac{d}{d x} [x - 2]$ .

$0 + 5 (\cos (\frac{π}{12} (x - 2)) (\frac{π}{12} \frac{d}{d x} [x - 2]))$

Step 2.2.4

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

$0 + 5 (\cos (\frac{π}{12} (x - 2)) (\frac{π}{12} (\frac{d}{d x} [x] + \frac{d}{d x} [- 2])))$

Step 2.2.5

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

$0 + 5 (\cos (\frac{π}{12} (x - 2)) (\frac{π}{12} (1 + \frac{d}{d x} [- 2])))$

Step 2.2.6

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$0 + 5 (\cos (\frac{π}{12} (x - 2)) (\frac{π}{12} (1 + 0)))$

Step 2.2.7

Add $1$ and $0$ .

$0 + 5 (\cos (\frac{π}{12} (x - 2)) (\frac{π}{12} \cdot 1))$

Step 2.2.8

Multiply $\frac{π}{12}$ by $1$ .

$0 + 5 (\cos (\frac{π}{12} (x - 2)) \frac{π}{12})$

Step 2.2.9

Combine $\cos (\frac{π}{12} (x - 2))$ and $\frac{π}{12}$ .

$0 + 5 \frac{\cos (\frac{π}{12} (x - 2)) π}{12}$

Step 2.2.10

Combine $5$ and $\frac{\cos (\frac{π}{12} (x - 2)) π}{12}$ .

$0 + \frac{5 \cos (\frac{π}{12} (x - 2)) π}{12}$

Step 2.3

Simplify.

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Step 2.3.1

Apply the distributive property.

$0 + \frac{5 \cos (\frac{π}{12} x + \frac{π}{12} \cdot - 2) π}{12}$

Step 2.3.2

Combine terms.

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Step 2.3.2.1

Combine $\frac{π}{12}$ and $- 2$ .

$0 + \frac{5 \cos (\frac{π}{12} x + \frac{π \cdot - 2}{12}) π}{12}$

Step 2.3.2.2

Move $- 2$ to the left of $π$ .

$0 + \frac{5 \cos (\frac{π}{12} x + \frac{- 2 \cdot π}{12}) π}{12}$

Step 2.3.2.3

Cancel the common factor of $- 2$ and $12$ .

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Step 2.3.2.3.1

Factor $2$ out of $- 2 π$ .

$0 + \frac{5 \cos (\frac{π}{12} x + \frac{2 (- π)}{12}) π}{12}$

Step 2.3.2.3.2

Cancel the common factors.

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Step 2.3.2.3.2.1

Factor $2$ out of $12$ .

$0 + \frac{5 \cos (\frac{π}{12} x + \frac{2 (- π)}{2 (6)}) π}{12}$

Step 2.3.2.3.2.2

Cancel the common factor.

$0 + \frac{5 \cos (\frac{π}{12} x + \frac{2 (- π)}{2 \cdot 6}) π}{12}$

Step 2.3.2.3.2.3

Rewrite the expression.

$0 + \frac{5 \cos (\frac{π}{12} x + \frac{- π}{6}) π}{12}$

Step 2.3.2.4

Move the negative in front of the fraction.

$0 + \frac{5 \cos (\frac{π}{12} x - \frac{π}{6}) π}{12}$

Step 2.3.2.5

Combine $\frac{π}{12}$ and $x$ .

$0 + \frac{5 \cos (\frac{π x}{12} - \frac{π}{6}) π}{12}$

Step 2.3.2.6

Add $0$ and $\frac{5 \cos (\frac{π x}{12} - \frac{π}{6}) π}{12}$ .

$\frac{5 \cos (\frac{π x}{12} - \frac{π}{6}) π}{12}$

Step 2.3.3

Reorder terms.

$\frac{5 π \cos (\frac{π x}{12} - \frac{π}{6})}{12}$

Step 3

Find the second derivative of the function.

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Step 3.1

Since $\frac{5 π}{12}$ is constant with respect to $x$ , the derivative of $\frac{5 π \cos (\frac{π x}{12} - \frac{π}{6})}{12}$ with respect to $x$ is $\frac{5 π}{12} \frac{d}{d x} [\cos (\frac{π x}{12} - \frac{π}{6})]$ .

$f'' (x) = \frac{5 π}{12} \cdot \frac{d}{d x} (\cos (\frac{π x}{12} - \frac{π}{6}))$

Step 3.2

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 $g (x) = \frac{π x}{12} - \frac{π}{6}$ .

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Step 3.2.1

To apply the Chain Rule, set $u$ as $\frac{π x}{12} - \frac{π}{6}$ .

$f'' (x) = \frac{5 π}{12} \cdot (\frac{d}{d u} (\cos (u)) \frac{d}{d x} (\frac{π x}{12} - \frac{π}{6}))$

Step 3.2.2

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

$f'' (x) = \frac{5 π}{12} \cdot (- \sin (u) \frac{d}{d x} (\frac{π x}{12} - \frac{π}{6}))$

Step 3.2.3

Replace all occurrences of $u$ with $\frac{π x}{12} - \frac{π}{6}$ .

$f'' (x) = \frac{5 π}{12} \cdot (- \sin (\frac{π x}{12} - \frac{π}{6}) \frac{d}{d x} (\frac{π x}{12} - \frac{π}{6}))$

Step 3.3

Differentiate.

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Step 3.3.1

Combine $\frac{5 π}{12}$ and $\sin (\frac{π x}{12} - \frac{π}{6})$ .

$f'' (x) = - \frac{5 π \sin (\frac{π x}{12} - \frac{π}{6})}{12} \cdot \frac{d}{d x} (\frac{π x}{12} - \frac{π}{6})$

Step 3.3.2

By the Sum Rule, the derivative of $\frac{π x}{12} - \frac{π}{6}$ with respect to $x$ is $\frac{d}{d x} [\frac{π x}{12}] + \frac{d}{d x} [- \frac{π}{6}]$ .

$f'' (x) = - \frac{5 π \sin (\frac{π x}{12} - \frac{π}{6})}{12} \cdot (\frac{d}{d x} (\frac{π x}{12}) + \frac{d}{d x} (- \frac{π}{6}))$

Step 3.3.3

Since $\frac{π}{12}$ is constant with respect to $x$ , the derivative of $\frac{π x}{12}$ with respect to $x$ is $\frac{π}{12} \frac{d}{d x} [x]$ .

$f'' (x) = - \frac{5 π \sin (\frac{π x}{12} - \frac{π}{6})}{12} \cdot (\frac{π}{12} \cdot \frac{d}{d x} (x) + \frac{d}{d x} (- \frac{π}{6}))$

Step 3.3.4

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

$f'' (x) = - \frac{5 π \sin (\frac{π x}{12} - \frac{π}{6})}{12} \cdot (\frac{π}{12} \cdot 1 + \frac{d}{d x} (- \frac{π}{6}))$

Step 3.3.5

Multiply $\frac{π}{12}$ by $1$ .

$f'' (x) = - \frac{5 π \sin (\frac{π x}{12} - \frac{π}{6})}{12} \cdot (\frac{π}{12} + \frac{d}{d x} (- \frac{π}{6}))$

Step 3.3.6

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

$f'' (x) = - \frac{5 π \sin (\frac{π x}{12} - \frac{π}{6})}{12} \cdot (\frac{π}{12} + 0)$

Step 3.3.7

Combine fractions.

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Step 3.3.7.1

Add $\frac{π}{12}$ and $0$ .

$f'' (x) = - \frac{5 π \sin (\frac{π x}{12} - \frac{π}{6})}{12} \cdot \frac{π}{12}$

Step 3.3.7.2

Multiply $\frac{π}{12}$ by $\frac{5 π \sin (\frac{π x}{12} - \frac{π}{6})}{12}$ .

$f'' (x) = - \frac{π (5 π \sin (\frac{π x}{12} - \frac{π}{6}))}{12 \cdot 12}$

Step 3.4

Raise $π$ to the power of $1$ .

$f'' (x) = - \frac{5 (π π) \sin (\frac{π x}{12} - \frac{π}{6})}{12 \cdot 12}$

Step 3.5

Raise $π$ to the power of $1$ .

$f'' (x) = - \frac{5 (π π) \sin (\frac{π x}{12} - \frac{π}{6})}{12 \cdot 12}$

Step 3.6

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

$f'' (x) = - \frac{5 π^{1 + 1} \sin (\frac{π x}{12} - \frac{π}{6})}{12 \cdot 12}$

Step 3.7

Add $1$ and $1$ .

$f'' (x) = - \frac{5 π^{2} \sin (\frac{π x}{12} - \frac{π}{6})}{12 \cdot 12}$

Step 3.8

Multiply $12$ by $12$ .

$f'' (x) = - \frac{5 π^{2} \sin (\frac{π x}{12} - \frac{π}{6})}{144}$

Step 4

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

$\frac{5 π \cos (\frac{π x}{12} - \frac{π}{6})}{12} = 0$

Step 5

Set the numerator equal to zero.

$5 π \cos (\frac{π x}{12} - \frac{π}{6}) = 0$

Step 6

Solve the equation for $x$ .

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Step 6.1

Divide each term in $5 π \cos (\frac{π x}{12} - \frac{π}{6}) = 0$ by $5 π$ and simplify.

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Step 6.1.1

Divide each term in $5 π \cos (\frac{π x}{12} - \frac{π}{6}) = 0$ by $5 π$ .

$\frac{5 π \cos (\frac{π x}{12} - \frac{π}{6})}{5 π} = \frac{0}{5 π}$

Step 6.1.2

Simplify the left side.

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Step 6.1.2.1

Cancel the common factor of $5$ .

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Step 6.1.2.1.1

Cancel the common factor.

$\frac{5 π \cos (\frac{π x}{12} - \frac{π}{6})}{5 π} = \frac{0}{5 π}$

Step 6.1.2.1.2

Rewrite the expression.

$\frac{π \cos (\frac{π x}{12} - \frac{π}{6})}{π} = \frac{0}{5 π}$

Step 6.1.2.2

Cancel the common factor of $π$ .

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Step 6.1.2.2.1

Cancel the common factor.

$\frac{π \cos (\frac{π x}{12} - \frac{π}{6})}{π} = \frac{0}{5 π}$

Step 6.1.2.2.2

Divide $\cos (\frac{π x}{12} - \frac{π}{6})$ by $1$ .

$\cos (\frac{π x}{12} - \frac{π}{6}) = \frac{0}{5 π}$

Step 6.1.3

Simplify the right side.

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Step 6.1.3.1

Cancel the common factor of $0$ and $5$ .

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Step 6.1.3.1.1

Factor $5$ out of $0$ .

$\cos (\frac{π x}{12} - \frac{π}{6}) = \frac{5 (0)}{5 π}$

Step 6.1.3.1.2

Cancel the common factors.

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Step 6.1.3.1.2.1

Factor $5$ out of $5 π$ .

$\cos (\frac{π x}{12} - \frac{π}{6}) = \frac{5 (0)}{5 (π)}$

Step 6.1.3.1.2.2

Cancel the common factor.

$\cos (\frac{π x}{12} - \frac{π}{6}) = \frac{5 \cdot 0}{5 π}$

Step 6.1.3.1.2.3

Rewrite the expression.

$\cos (\frac{π x}{12} - \frac{π}{6}) = \frac{0}{π}$

Step 6.1.3.2

Divide $0$ by $π$ .

$\cos (\frac{π x}{12} - \frac{π}{6}) = 0$

Step 6.2

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

$\frac{π x}{12} - \frac{π}{6} = \arccos (0)$

Step 6.3

Simplify the right side.

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Step 6.3.1

The exact value of $\arccos (0)$ is $\frac{π}{2}$ .

$\frac{π x}{12} - \frac{π}{6} = \frac{π}{2}$

Step 6.4

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

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Step 6.4.1

Add $\frac{π}{6}$ to both sides of the equation.

$\frac{π x}{12} = \frac{π}{2} + \frac{π}{6}$

Step 6.4.2

To write $\frac{π}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{π x}{12} = \frac{π}{2} \cdot \frac{3}{3} + \frac{π}{6}$

Step 6.4.3

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.4.3.1

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

$\frac{π x}{12} = \frac{π \cdot 3}{2 \cdot 3} + \frac{π}{6}$

Step 6.4.3.2

Multiply $2$ by $3$ .

$\frac{π x}{12} = \frac{π \cdot 3}{6} + \frac{π}{6}$

Step 6.4.4

Combine the numerators over the common denominator.

$\frac{π x}{12} = \frac{π \cdot 3 + π}{6}$

Step 6.4.5

Simplify the numerator.

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Step 6.4.5.1

Move $3$ to the left of $π$ .

$\frac{π x}{12} = \frac{3 \cdot π + π}{6}$

Step 6.4.5.2

Add $3 π$ and $π$ .

$\frac{π x}{12} = \frac{4 π}{6}$

Step 6.4.6

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

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Step 6.4.6.1

Factor $2$ out of $4 π$ .

$\frac{π x}{12} = \frac{2 (2 π)}{6}$

Step 6.4.6.2

Cancel the common factors.

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Step 6.4.6.2.1

Factor $2$ out of $6$ .

$\frac{π x}{12} = \frac{2 (2 π)}{2 (3)}$

Step 6.4.6.2.2

Cancel the common factor.

$\frac{π x}{12} = \frac{2 (2 π)}{2 \cdot 3}$

Step 6.4.6.2.3

Rewrite the expression.

$\frac{π x}{12} = \frac{2 π}{3}$

Step 6.5

Multiply both sides of the equation by $\frac{12}{π}$ .

$\frac{12}{π} \cdot \frac{π x}{12} = \frac{12}{π} \cdot \frac{2 π}{3}$

Step 6.6

Simplify both sides of the equation.

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Step 6.6.1

Simplify the left side.

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Step 6.6.1.1

Simplify $\frac{12}{π} \cdot \frac{π x}{12}$ .

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Step 6.6.1.1.1

Cancel the common factor of $12$ .

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Step 6.6.1.1.1.1

Cancel the common factor.

$\frac{12}{π} \cdot \frac{π x}{12} = \frac{12}{π} \cdot \frac{2 π}{3}$

Step 6.6.1.1.1.2

Rewrite the expression.

$\frac{1}{π} (π x) = \frac{12}{π} \cdot \frac{2 π}{3}$

Step 6.6.1.1.2

Cancel the common factor of $π$ .

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Step 6.6.1.1.2.1

Factor $π$ out of $π x$ .

$\frac{1}{π} (π (x)) = \frac{12}{π} \cdot \frac{2 π}{3}$

Step 6.6.1.1.2.2

Cancel the common factor.

$\frac{1}{π} (π x) = \frac{12}{π} \cdot \frac{2 π}{3}$

Step 6.6.1.1.2.3

Rewrite the expression.

$x = \frac{12}{π} \cdot \frac{2 π}{3}$

Step 6.6.2

Simplify the right side.

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Step 6.6.2.1

Simplify $\frac{12}{π} \cdot \frac{2 π}{3}$ .

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Step 6.6.2.1.1

Cancel the common factor of $3$ .

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Step 6.6.2.1.1.1

Factor $3$ out of $12$ .

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

Step 6.6.2.1.1.2

Cancel the common factor.

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

Step 6.6.2.1.1.3

Rewrite the expression.

$x = \frac{4}{π} (2 π)$

Step 6.6.2.1.2

Cancel the common factor of $π$ .

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Step 6.6.2.1.2.1

Factor $π$ out of $2 π$ .

$x = \frac{4}{π} (π \cdot 2)$

Step 6.6.2.1.2.2

Cancel the common factor.

$x = \frac{4}{π} (π \cdot 2)$

Step 6.6.2.1.2.3

Rewrite the expression.

$x = 4 \cdot 2$

Step 6.6.2.1.3

Multiply $4$ by $2$ .

$x = 8$

Step 6.7

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the fourth quadrant.

$\frac{π x}{12} - \frac{π}{6} = 2 π - \frac{π}{2}$

Step 6.8

Solve for $x$ .

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Step 6.8.1

Simplify $2 π - \frac{π}{2}$ .

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Step 6.8.1.1

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

$\frac{π x}{12} - \frac{π}{6} = 2 π \cdot \frac{2}{2} - \frac{π}{2}$

Step 6.8.1.2

Combine fractions.

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Step 6.8.1.2.1

Combine $2 π$ and $\frac{2}{2}$ .

$\frac{π x}{12} - \frac{π}{6} = \frac{2 π \cdot 2}{2} - \frac{π}{2}$

Step 6.8.1.2.2

Combine the numerators over the common denominator.

$\frac{π x}{12} - \frac{π}{6} = \frac{2 π \cdot 2 - π}{2}$

Step 6.8.1.3

Simplify the numerator.

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Step 6.8.1.3.1

Multiply $2$ by $2$ .

$\frac{π x}{12} - \frac{π}{6} = \frac{4 π - π}{2}$

Step 6.8.1.3.2

Subtract $π$ from $4 π$ .

$\frac{π x}{12} - \frac{π}{6} = \frac{3 π}{2}$

Step 6.8.2

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

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Step 6.8.2.1

Add $\frac{π}{6}$ to both sides of the equation.

$\frac{π x}{12} = \frac{3 π}{2} + \frac{π}{6}$

Step 6.8.2.2

To write $\frac{3 π}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{π x}{12} = \frac{3 π}{2} \cdot \frac{3}{3} + \frac{π}{6}$

Step 6.8.2.3

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.8.2.3.1

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

$\frac{π x}{12} = \frac{3 π \cdot 3}{2 \cdot 3} + \frac{π}{6}$

Step 6.8.2.3.2

Multiply $2$ by $3$ .

$\frac{π x}{12} = \frac{3 π \cdot 3}{6} + \frac{π}{6}$

Step 6.8.2.4

Combine the numerators over the common denominator.

$\frac{π x}{12} = \frac{3 π \cdot 3 + π}{6}$

Step 6.8.2.5

Simplify the numerator.

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Step 6.8.2.5.1

Multiply $3$ by $3$ .

$\frac{π x}{12} = \frac{9 π + π}{6}$

Step 6.8.2.5.2

Add $9 π$ and $π$ .

$\frac{π x}{12} = \frac{10 π}{6}$

Step 6.8.2.6

Cancel the common factor of $10$ and $6$ .

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Step 6.8.2.6.1

Factor $2$ out of $10 π$ .

$\frac{π x}{12} = \frac{2 (5 π)}{6}$

Step 6.8.2.6.2

Cancel the common factors.

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Step 6.8.2.6.2.1

Factor $2$ out of $6$ .

$\frac{π x}{12} = \frac{2 (5 π)}{2 (3)}$

Step 6.8.2.6.2.2

Cancel the common factor.

$\frac{π x}{12} = \frac{2 (5 π)}{2 \cdot 3}$

Step 6.8.2.6.2.3

Rewrite the expression.

$\frac{π x}{12} = \frac{5 π}{3}$

Step 6.8.3

Multiply both sides of the equation by $\frac{12}{π}$ .

$\frac{12}{π} \cdot \frac{π x}{12} = \frac{12}{π} \cdot \frac{5 π}{3}$

Step 6.8.4

Simplify both sides of the equation.

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Step 6.8.4.1

Simplify the left side.

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Step 6.8.4.1.1

Simplify $\frac{12}{π} \cdot \frac{π x}{12}$ .

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Step 6.8.4.1.1.1

Cancel the common factor of $12$ .

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Step 6.8.4.1.1.1.1

Cancel the common factor.

$\frac{12}{π} \cdot \frac{π x}{12} = \frac{12}{π} \cdot \frac{5 π}{3}$

Step 6.8.4.1.1.1.2

Rewrite the expression.

$\frac{1}{π} (π x) = \frac{12}{π} \cdot \frac{5 π}{3}$

Step 6.8.4.1.1.2

Cancel the common factor of $π$ .

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Step 6.8.4.1.1.2.1

Factor $π$ out of $π x$ .

$\frac{1}{π} (π (x)) = \frac{12}{π} \cdot \frac{5 π}{3}$

Step 6.8.4.1.1.2.2

Cancel the common factor.

$\frac{1}{π} (π x) = \frac{12}{π} \cdot \frac{5 π}{3}$

Step 6.8.4.1.1.2.3

Rewrite the expression.

$x = \frac{12}{π} \cdot \frac{5 π}{3}$

Step 6.8.4.2

Simplify the right side.

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Step 6.8.4.2.1

Simplify $\frac{12}{π} \cdot \frac{5 π}{3}$ .

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Step 6.8.4.2.1.1

Cancel the common factor of $3$ .

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Step 6.8.4.2.1.1.1

Factor $3$ out of $12$ .

$x = \frac{3 (4)}{π} \cdot \frac{5 π}{3}$

Step 6.8.4.2.1.1.2

Cancel the common factor.

$x = \frac{3 \cdot 4}{π} \cdot \frac{5 π}{3}$

Step 6.8.4.2.1.1.3

Rewrite the expression.

$x = \frac{4}{π} (5 π)$

Step 6.8.4.2.1.2

Cancel the common factor of $π$ .

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Step 6.8.4.2.1.2.1

Factor $π$ out of $5 π$ .

$x = \frac{4}{π} (π \cdot 5)$

Step 6.8.4.2.1.2.2

Cancel the common factor.

$x = \frac{4}{π} (π \cdot 5)$

Step 6.8.4.2.1.2.3

Rewrite the expression.

$x = 4 \cdot 5$

Step 6.8.4.2.1.3

Multiply $4$ by $5$ .

$x = 20$

Step 6.9

The solution to the equation $\frac{π x}{12} - \frac{π}{6} = \frac{π}{2}$ .

$x = 8, 20$

Step 7

Evaluate the second derivative at $x = 8$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- \frac{5 π^{2} \sin (\frac{π (8)}{12} - \frac{π}{6})}{144}$

Step 8

Evaluate the second derivative.

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Step 8.1

Cancel the common factor of $8$ and $12$ .

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Step 8.1.1

Factor $4$ out of $π (8)$ .

$- \frac{5 π^{2} \sin (\frac{4 (π \cdot (2))}{12} - \frac{π}{6})}{144}$

Step 8.1.2

Cancel the common factors.

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Step 8.1.2.1

Factor $4$ out of $12$ .

$- \frac{5 π^{2} \sin (\frac{4 (π \cdot (2))}{4 (3)} - \frac{π}{6})}{144}$

Step 8.1.2.2

Cancel the common factor.

$- \frac{5 π^{2} \sin (\frac{4 (π \cdot (2))}{4 \cdot 3} - \frac{π}{6})}{144}$

Step 8.1.2.3

Rewrite the expression.

$- \frac{5 π^{2} \sin (\frac{π \cdot (2)}{3} - \frac{π}{6})}{144}$

Step 8.2

Simplify the numerator.

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Step 8.2.1

Move $2$ to the left of $π$ .

$- \frac{5 π^{2} \sin (\frac{2 π}{3} - \frac{π}{6})}{144}$

Step 8.2.2

To write $\frac{2 π}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{5 π^{2} \sin (\frac{2 π}{3} \cdot \frac{2}{2} - \frac{π}{6})}{144}$

Step 8.2.3

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.2.3.1

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

$- \frac{5 π^{2} \sin (\frac{2 π \cdot 2}{3 \cdot 2} - \frac{π}{6})}{144}$

Step 8.2.3.2

Multiply $3$ by $2$ .

$- \frac{5 π^{2} \sin (\frac{2 π \cdot 2}{6} - \frac{π}{6})}{144}$

Step 8.2.4

Combine the numerators over the common denominator.

$- \frac{5 π^{2} \sin (\frac{2 π \cdot 2 - π}{6})}{144}$

Step 8.2.5

Simplify the numerator.

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Step 8.2.5.1

Multiply $2$ by $2$ .

$- \frac{5 π^{2} \sin (\frac{4 π - π}{6})}{144}$

Step 8.2.5.2

Subtract $π$ from $4 π$ .

$- \frac{5 π^{2} \sin (\frac{3 π}{6})}{144}$

Step 8.2.6

Cancel the common factor of $3$ and $6$ .

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Step 8.2.6.1

Factor $3$ out of $3 π$ .

$- \frac{5 π^{2} \sin (\frac{3 (π)}{6})}{144}$

Step 8.2.6.2

Cancel the common factors.

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Step 8.2.6.2.1

Factor $3$ out of $6$ .

$- \frac{5 π^{2} \sin (\frac{3 π}{3 \cdot 2})}{144}$

Step 8.2.6.2.2

Cancel the common factor.

$- \frac{5 π^{2} \sin (\frac{3 π}{3 \cdot 2})}{144}$

Step 8.2.6.2.3

Rewrite the expression.

$- \frac{5 π^{2} \sin (\frac{π}{2})}{144}$

Step 8.2.7

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$- \frac{5 π^{2} \cdot 1}{144}$

Step 8.2.8

Multiply $5$ by $1$ .

$- \frac{5 π^{2}}{144}$

Step 9

$x = 8$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = 8$ is a local maximum

Step 10

Find the y-value when $x = 8$ .

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Step 10.1

Replace the variable $x$ with $8$ in the expression.

$f (8) = - 2 + 5 \sin (\frac{π}{12} \cdot ((8) - 2))$

Step 10.2

Simplify the result.

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Step 10.2.1

Simplify each term.

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Step 10.2.1.1

Subtract $2$ from $8$ .

$f (8) = - 2 + 5 \sin (\frac{π}{12} \cdot 6)$

Step 10.2.1.2

Cancel the common factor of $6$ .

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Step 10.2.1.2.1

Factor $6$ out of $12$ .

$f (8) = - 2 + 5 \sin (\frac{π}{6 (2)} \cdot 6)$

Step 10.2.1.2.2

Cancel the common factor.

$f (8) = - 2 + 5 \sin (\frac{π}{6 \cdot 2} \cdot 6)$

Step 10.2.1.2.3

Rewrite the expression.

$f (8) = - 2 + 5 \sin (\frac{π}{2})$

Step 10.2.1.3

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$f (8) = - 2 + 5 \cdot 1$

Step 10.2.1.4

Multiply $5$ by $1$ .

$f (8) = - 2 + 5$

Step 10.2.2

Add $- 2$ and $5$ .

$f (8) = 3$

Step 10.2.3

The final answer is $3$ .

$y = 3$

Step 11

Evaluate the second derivative at $x = 20$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- \frac{5 π^{2} \sin (\frac{π (20)}{12} - \frac{π}{6})}{144}$

Step 12

Evaluate the second derivative.

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Step 12.1

Cancel the common factor of $20$ and $12$ .

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Step 12.1.1

Factor $4$ out of $π (20)$ .

$- \frac{5 π^{2} \sin (\frac{4 (π \cdot (5))}{12} - \frac{π}{6})}{144}$

Step 12.1.2

Cancel the common factors.

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Step 12.1.2.1

Factor $4$ out of $12$ .

$- \frac{5 π^{2} \sin (\frac{4 (π \cdot (5))}{4 (3)} - \frac{π}{6})}{144}$

Step 12.1.2.2

Cancel the common factor.

$- \frac{5 π^{2} \sin (\frac{4 (π \cdot (5))}{4 \cdot 3} - \frac{π}{6})}{144}$

Step 12.1.2.3

Rewrite the expression.

$- \frac{5 π^{2} \sin (\frac{π \cdot (5)}{3} - \frac{π}{6})}{144}$

Step 12.2

Simplify the numerator.

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Step 12.2.1

Move $5$ to the left of $π$ .

$- \frac{5 π^{2} \sin (\frac{5 π}{3} - \frac{π}{6})}{144}$

Step 12.2.2

To write $\frac{5 π}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{5 π^{2} \sin (\frac{5 π}{3} \cdot \frac{2}{2} - \frac{π}{6})}{144}$

Step 12.2.3

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 12.2.3.1

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

$- \frac{5 π^{2} \sin (\frac{5 π \cdot 2}{3 \cdot 2} - \frac{π}{6})}{144}$

Step 12.2.3.2

Multiply $3$ by $2$ .

$- \frac{5 π^{2} \sin (\frac{5 π \cdot 2}{6} - \frac{π}{6})}{144}$

Step 12.2.4

Combine the numerators over the common denominator.

$- \frac{5 π^{2} \sin (\frac{5 π \cdot 2 - π}{6})}{144}$

Step 12.2.5

Simplify the numerator.

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Step 12.2.5.1

Multiply $2$ by $5$ .

$- \frac{5 π^{2} \sin (\frac{10 π - π}{6})}{144}$

Step 12.2.5.2

Subtract $π$ from $10 π$ .

$- \frac{5 π^{2} \sin (\frac{9 π}{6})}{144}$

Step 12.2.6

Cancel the common factor of $9$ and $6$ .

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Step 12.2.6.1

Factor $3$ out of $9 π$ .

$- \frac{5 π^{2} \sin (\frac{3 (3 π)}{6})}{144}$

Step 12.2.6.2

Cancel the common factors.

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Step 12.2.6.2.1

Factor $3$ out of $6$ .

$- \frac{5 π^{2} \sin (\frac{3 (3 π)}{3 (2)})}{144}$

Step 12.2.6.2.2

Cancel the common factor.

$- \frac{5 π^{2} \sin (\frac{3 (3 π)}{3 \cdot 2})}{144}$

Step 12.2.6.2.3

Rewrite the expression.

$- \frac{5 π^{2} \sin (\frac{3 π}{2})}{144}$

Step 12.2.7

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

$- \frac{5 π^{2} (- \sin (\frac{π}{2}))}{144}$

Step 12.2.8

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$- \frac{5 π^{2} (- 1 \cdot 1)}{144}$

Step 12.2.9

Multiply $- 1$ by $1$ .

$- \frac{5 π^{2} \cdot - 1}{144}$

Step 12.2.10

Multiply $- 1$ by $5$ .

$- \frac{- 5 π^{2}}{144}$

Step 12.3

Move the negative in front of the fraction.

$- - \frac{5 π^{2}}{144}$

Step 12.4

Multiply $- - \frac{5 π^{2}}{144}$ .

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Step 12.4.1

Multiply $- 1$ by $- 1$ .

$1 \frac{5 π^{2}}{144}$

Step 12.4.2

Multiply $\frac{5 π^{2}}{144}$ by $1$ .

$\frac{5 π^{2}}{144}$

Step 13

$x = 20$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = 20$ is a local minimum

Step 14

Find the y-value when $x = 20$ .

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Step 14.1

Replace the variable $x$ with $20$ in the expression.

$f (20) = - 2 + 5 \sin (\frac{π}{12} \cdot ((20) - 2))$

Step 14.2

Simplify the result.

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Step 14.2.1

Simplify each term.

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Step 14.2.1.1

Subtract $2$ from $20$ .

$f (20) = - 2 + 5 \sin (\frac{π}{12} \cdot 18)$

Step 14.2.1.2

Cancel the common factor of $6$ .

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Step 14.2.1.2.1

Factor $6$ out of $12$ .

$f (20) = - 2 + 5 \sin (\frac{π}{6 (2)} \cdot 18)$

Step 14.2.1.2.2

Factor $6$ out of $18$ .

$f (20) = - 2 + 5 \sin (\frac{π}{6 \cdot 2} \cdot (6 \cdot 3))$

Step 14.2.1.2.3

Cancel the common factor.

$f (20) = - 2 + 5 \sin (\frac{π}{6 \cdot 2} \cdot (6 \cdot 3))$

Step 14.2.1.2.4

Rewrite the expression.

$f (20) = - 2 + 5 \sin (\frac{π}{2} \cdot 3)$

Step 14.2.1.3

Combine $\frac{π}{2}$ and $3$ .

$f (20) = - 2 + 5 \sin (\frac{π \cdot 3}{2})$

Step 14.2.1.4

Move $3$ to the left of $π$ .

$f (20) = - 2 + 5 \sin (\frac{3 \cdot π}{2})$

Step 14.2.1.5

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

$f (20) = - 2 + 5 (- \sin (\frac{π}{2}))$

Step 14.2.1.6

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$f (20) = - 2 + 5 (- 1 \cdot 1)$

Step 14.2.1.7

Multiply $5 (- 1 \cdot 1)$ .

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Step 14.2.1.7.1

Multiply $- 1$ by $1$ .

$f (20) = - 2 + 5 \cdot - 1$

Step 14.2.1.7.2

Multiply $5$ by $- 1$ .

$f (20) = - 2 - 5$

Step 14.2.2

Subtract $5$ from $- 2$ .

$f (20) = - 7$

Step 14.2.3

The final answer is $- 7$ .

$y = - 7$

Step 15

These are the local extrema for $f (x) = - 2 + 5 \sin (\frac{π}{12} \cdot (x - 2))$ .

$(8, 3)$ is a local maxima

$(20, - 7)$ is a local minima

Step 16