Calculus Examples

$y = - 2 - 4 \sin (\frac{π}{2} (x - \frac{1}{3}))$

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 $- 2 - 4 \sin (\frac{π}{2} \cdot (x - \frac{1}{3}))$ with respect to $x$ is $\frac{d}{d x} [- 2] + \frac{d}{d x} [- 4 \sin (\frac{π}{2} \cdot (x - \frac{1}{3}))]$ .

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

Step 1.1.2

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

$0 + \frac{d}{d x} [- 4 \sin (\frac{π}{2} \cdot (x - \frac{1}{3}))]$

Step 1.2

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

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Step 1.2.1

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

$0 - 4 \frac{d}{d x} [\sin (\frac{π}{2} (x - \frac{1}{3}))]$

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 $f (x) = \sin (x)$ and $g (x) = \frac{π}{2} (x - \frac{1}{3})$ .

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Step 1.2.2.1

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

$0 - 4 (\frac{d}{d u} [\sin (u)] \frac{d}{d x} [\frac{π}{2} (x - \frac{1}{3})])$

Step 1.2.2.2

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

$0 - 4 (\cos (\frac{π}{2} (x - \frac{1}{3})) \frac{d}{d x} [\frac{π}{2} (x - \frac{1}{3})])$

Step 1.2.3

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

$0 - 4 (\cos (\frac{π}{2} (x - \frac{1}{3})) (\frac{π}{2} \frac{d}{d x} [x - \frac{1}{3}]))$

Step 1.2.4

By the Sum Rule, the derivative of $x - \frac{1}{3}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- \frac{1}{3}]$ .

$0 - 4 (\cos (\frac{π}{2} (x - \frac{1}{3})) (\frac{π}{2} (\frac{d}{d x} [x] + \frac{d}{d x} [- \frac{1}{3}])))$

Step 1.2.5

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

$0 - 4 (\cos (\frac{π}{2} (x - \frac{1}{3})) (\frac{π}{2} (1 + \frac{d}{d x} [- \frac{1}{3}])))$

Step 1.2.6

Since $- \frac{1}{3}$ is constant with respect to $x$ , the derivative of $- \frac{1}{3}$ with respect to $x$ is $0$ .

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

Step 1.2.7

Add $1$ and $0$ .

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

Step 1.2.8

Multiply $\frac{π}{2}$ by $1$ .

$0 - 4 (\cos (\frac{π}{2} (x - \frac{1}{3})) \frac{π}{2})$

Step 1.2.9

Combine $\cos (\frac{π}{2} (x - \frac{1}{3}))$ and $\frac{π}{2}$ .

$0 - 4 \frac{\cos (\frac{π}{2} (x - \frac{1}{3})) π}{2}$

Step 1.2.10

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

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

Step 1.2.11

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

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Step 1.2.11.1

Factor $2$ out of $- 4 \cos (\frac{π}{2} (x - \frac{1}{3})) π$ .

$0 + \frac{2 (- 2 \cos (\frac{π}{2} (x - \frac{1}{3})) π)}{2}$

Step 1.2.11.2

Cancel the common factors.

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Step 1.2.11.2.1

Factor $2$ out of $2$ .

$0 + \frac{2 (- 2 \cos (\frac{π}{2} (x - \frac{1}{3})) π)}{2 (1)}$

Step 1.2.11.2.2

Cancel the common factor.

$0 + \frac{2 (- 2 \cos (\frac{π}{2} (x - \frac{1}{3})) π)}{2 \cdot 1}$

Step 1.2.11.2.3

Rewrite the expression.

$0 + \frac{- 2 \cos (\frac{π}{2} (x - \frac{1}{3})) π}{1}$

Step 1.2.11.2.4

Divide $- 2 \cos (\frac{π}{2} (x - \frac{1}{3})) π$ by $1$ .

$0 - 2 \cos (\frac{π}{2} (x - \frac{1}{3})) π$

Step 1.3

Simplify.

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Step 1.3.1

Apply the distributive property.

$0 - 2 \cos (\frac{π}{2} x + \frac{π}{2} (- \frac{1}{3})) π$

Step 1.3.2

Combine terms.

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Step 1.3.2.1

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

$0 - 2 \cos (\frac{π}{2} x - \frac{π}{2 \cdot 3}) π$

Step 1.3.2.2

Multiply $2$ by $3$ .

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

Step 1.3.2.3

Combine $\frac{π}{2}$ and $x$ .

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

Step 1.3.2.4

Subtract $2 \cos (\frac{π x}{2} - \frac{π}{6}) π$ from $0$ .

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

Step 1.3.3

Reorder the factors of $- 2 \cos (\frac{π x}{2} - \frac{π}{6}) π$ .

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

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 $- 2 π \cos (\frac{π x}{2} - \frac{π}{6})$ with respect to $x$ is $- 2 π \frac{d}{d x} [\cos (\frac{π x}{2} - \frac{π}{6})]$ .

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

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 $f (x) = \cos (x)$ and $g (x) = \frac{π x}{2} - \frac{π}{6}$ .

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.3

Differentiate.

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Step 2.3.1

Multiply $- 1$ by $- 2$ .

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.3.4

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

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

Step 2.3.5

Multiply $\frac{π}{2}$ by $1$ .

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

Step 2.3.6

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

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

Step 2.3.7

Combine fractions.

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Step 2.3.7.1

Add $\frac{π}{2}$ and $0$ .

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

Step 2.3.7.2

Combine $\frac{π}{2}$ and $2$ .

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

Step 2.3.7.3

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

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

Step 2.4

Raise $π$ to the power of $1$ .

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

Step 2.5

Raise $π$ to the power of $1$ .

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

Step 2.6

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

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

Step 2.7

Add $1$ and $1$ .

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

Step 2.8

Combine $\frac{2 π^{2}}{2}$ and $\sin (\frac{π x}{2} - \frac{π}{6})$ .

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

Step 2.9

Cancel the common factor of $2$ .

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Step 2.9.1

Cancel the common factor.

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

Step 2.9.2

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

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

Step 3

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

$- 2 π \cos (\frac{π x}{2} - \frac{π}{6}) = 0$

Step 4

Divide each term in $- 2 π \cos (\frac{π x}{2} - \frac{π}{6}) = 0$ by $- 2 π$ and simplify.

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Step 4.1

Divide each term in $- 2 π \cos (\frac{π x}{2} - \frac{π}{6}) = 0$ by $- 2 π$ .

$\frac{- 2 π \cos (\frac{π x}{2} - \frac{π}{6})}{- 2 π} = \frac{0}{- 2 π}$

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.

$\frac{- 2 π \cos (\frac{π x}{2} - \frac{π}{6})}{- 2 π} = \frac{0}{- 2 π}$

Step 4.2.1.2

Rewrite the expression.

$\frac{π \cos (\frac{π x}{2} - \frac{π}{6})}{π} = \frac{0}{- 2 π}$

Step 4.2.2

Cancel the common factor of $π$ .

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Step 4.2.2.1

Cancel the common factor.

$\frac{π \cos (\frac{π x}{2} - \frac{π}{6})}{π} = \frac{0}{- 2 π}$

Step 4.2.2.2

Divide $\cos (\frac{π x}{2} - \frac{π}{6})$ by $1$ .

$\cos (\frac{π x}{2} - \frac{π}{6}) = \frac{0}{- 2 π}$

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 $0$ .

$\cos (\frac{π x}{2} - \frac{π}{6}) = \frac{2 (0)}{- 2 π}$

Step 4.3.1.2

Cancel the common factors.

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Step 4.3.1.2.1

Factor $2$ out of $- 2 π$ .

$\cos (\frac{π x}{2} - \frac{π}{6}) = \frac{2 (0)}{2 (- π)}$

Step 4.3.1.2.2

Cancel the common factor.

$\cos (\frac{π x}{2} - \frac{π}{6}) = \frac{2 \cdot 0}{2 (- π)}$

Step 4.3.1.2.3

Rewrite the expression.

$\cos (\frac{π x}{2} - \frac{π}{6}) = \frac{0}{- π}$

Step 4.3.2

Divide $0$ by $- π$ .

$\cos (\frac{π x}{2} - \frac{π}{6}) = 0$

Step 5

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

$\frac{π x}{2} - \frac{π}{6} = \arccos (0)$

Step 6

Simplify the right side.

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Step 6.1

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

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

Step 7

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

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

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

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Step 7.3.1

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

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

Step 7.3.2

Multiply $2$ by $3$ .

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

Step 7.4

Combine the numerators over the common denominator.

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

Step 7.5

Simplify the numerator.

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Step 7.5.1

Move $3$ to the left of $π$ .

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

Step 7.5.2

Add $3 π$ and $π$ .

$\frac{π x}{2} = \frac{4 π}{6}$

Step 7.6

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

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Step 7.6.1

Factor $2$ out of $4 π$ .

$\frac{π x}{2} = \frac{2 (2 π)}{6}$

Step 7.6.2

Cancel the common factors.

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Step 7.6.2.1

Factor $2$ out of $6$ .

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

Step 7.6.2.2

Cancel the common factor.

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

Step 7.6.2.3

Rewrite the expression.

$\frac{π x}{2} = \frac{2 π}{3}$

Step 8

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

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

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 $\frac{2}{π} \cdot \frac{π x}{2}$ .

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Step 9.1.1.1

Cancel the common factor of $2$ .

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Step 9.1.1.1.1

Cancel the common factor.

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

Step 9.1.1.1.2

Rewrite the expression.

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

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 $π x$ .

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

Step 9.1.1.2.2

Cancel the common factor.

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

Step 9.1.1.2.3

Rewrite the expression.

$x = \frac{2}{π} \cdot \frac{2 π}{3}$

Step 9.2

Simplify the right side.

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Step 9.2.1

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

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Step 9.2.1.1

Cancel the common factor of $π$ .

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Step 9.2.1.1.1

Factor $π$ out of $2 π$ .

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

Step 9.2.1.1.2

Cancel the common factor.

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

Step 9.2.1.1.3

Rewrite the expression.

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

Step 9.2.1.2

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

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

Step 9.2.1.3

Multiply $2$ by $2$ .

$x = \frac{4}{3}$

Step 10

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.

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

Step 11

Solve for $x$ .

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Step 11.1

Simplify $2 π - \frac{π}{2}$ .

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Step 11.1.1

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

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

Step 11.1.2

Combine fractions.

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Step 11.1.2.1

Combine $2 π$ and $\frac{2}{2}$ .

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

Step 11.1.2.2

Combine the numerators over the common denominator.

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

Step 11.1.3

Simplify the numerator.

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Step 11.1.3.1

Multiply $2$ by $2$ .

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

Step 11.1.3.2

Subtract $π$ from $4 π$ .

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

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

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

Step 11.2.2

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

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

Step 11.2.3

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

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Step 11.2.3.1

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

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

Step 11.2.3.2

Multiply $2$ by $3$ .

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

Step 11.2.4

Combine the numerators over the common denominator.

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

Step 11.2.5

Simplify the numerator.

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Step 11.2.5.1

Multiply $3$ by $3$ .

$\frac{π x}{2} = \frac{9 π + π}{6}$

Step 11.2.5.2

Add $9 π$ and $π$ .

$\frac{π x}{2} = \frac{10 π}{6}$

Step 11.2.6

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

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Step 11.2.6.1

Factor $2$ out of $10 π$ .

$\frac{π x}{2} = \frac{2 (5 π)}{6}$

Step 11.2.6.2

Cancel the common factors.

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Step 11.2.6.2.1

Factor $2$ out of $6$ .

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

Step 11.2.6.2.2

Cancel the common factor.

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

Step 11.2.6.2.3

Rewrite the expression.

$\frac{π x}{2} = \frac{5 π}{3}$

Step 11.3

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

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

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 $\frac{2}{π} \cdot \frac{π x}{2}$ .

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Step 11.4.1.1.1

Cancel the common factor of $2$ .

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Step 11.4.1.1.1.1

Cancel the common factor.

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

Step 11.4.1.1.1.2

Rewrite the expression.

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

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 $π x$ .

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

Step 11.4.1.1.2.2

Cancel the common factor.

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

Step 11.4.1.1.2.3

Rewrite the expression.

$x = \frac{2}{π} \cdot \frac{5 π}{3}$

Step 11.4.2

Simplify the right side.

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Step 11.4.2.1

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

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Step 11.4.2.1.1

Cancel the common factor of $π$ .

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Step 11.4.2.1.1.1

Factor $π$ out of $5 π$ .

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

Step 11.4.2.1.1.2

Cancel the common factor.

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

Step 11.4.2.1.1.3

Rewrite the expression.

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

Step 11.4.2.1.2

Combine $2$ and $\frac{5}{3}$ .

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

Step 11.4.2.1.3

Multiply $2$ by $5$ .

$x = \frac{10}{3}$

Step 12

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

$x = \frac{4}{3}, \frac{10}{3}$

Step 13

Evaluate the second derivative at $x = \frac{4}{3}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

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

Step 14

Evaluate the second derivative.

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Step 14.1

Simplify each term.

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Step 14.1.1

Combine $π$ and $\frac{4}{3}$ .

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

Step 14.1.2

Move $4$ to the left of $π$ .

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

Step 14.1.3

Multiply the numerator by the reciprocal of the denominator.

$π^{2} \sin (\frac{4 π}{3} \cdot \frac{1}{2} - \frac{π}{6})$

Step 14.1.4

Cancel the common factor of $2$ .

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Step 14.1.4.1

Factor $2$ out of $4 π$ .

$π^{2} \sin (\frac{2 (2 π)}{3} \cdot \frac{1}{2} - \frac{π}{6})$

Step 14.1.4.2

Cancel the common factor.

$π^{2} \sin (\frac{2 (2 π)}{3} \cdot \frac{1}{2} - \frac{π}{6})$

Step 14.1.4.3

Rewrite the expression.

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

Step 14.2

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

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

Step 14.3

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

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Step 14.3.1

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

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

Step 14.3.2

Multiply $3$ by $2$ .

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

Step 14.4

Combine the numerators over the common denominator.

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

Step 14.5

Simplify the numerator.

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Step 14.5.1

Multiply $2$ by $2$ .

$π^{2} \sin (\frac{4 π - π}{6})$

Step 14.5.2

Subtract $π$ from $4 π$ .

$π^{2} \sin (\frac{3 π}{6})$

Step 14.6

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

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Step 14.6.1

Factor $3$ out of $3 π$ .

$π^{2} \sin (\frac{3 (π)}{6})$

Step 14.6.2

Cancel the common factors.

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Step 14.6.2.1

Factor $3$ out of $6$ .

$π^{2} \sin (\frac{3 π}{3 \cdot 2})$

Step 14.6.2.2

Cancel the common factor.

$π^{2} \sin (\frac{3 π}{3 \cdot 2})$

Step 14.6.2.3

Rewrite the expression.

$π^{2} \sin (\frac{π}{2})$

Step 14.7

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

$π^{2} \cdot 1$

Step 14.8

Multiply $π^{2}$ by $1$ .

$π^{2}$

Step 15

$x = \frac{4}{3}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = \frac{4}{3}$ is a local minimum

Step 16

Find the y-value when $x = \frac{4}{3}$ .

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Step 16.1

Replace the variable $x$ with $\frac{4}{3}$ in the expression.

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

Step 16.2

Simplify the result.

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Step 16.2.1

Simplify each term.

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Step 16.2.1.1

Combine the numerators over the common denominator.

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

Step 16.2.1.2

Subtract $1$ from $4$ .

$f (\frac{4}{3}) = - 2 - 4 \sin (\frac{π}{2} \cdot \frac{3}{3})$

Step 16.2.1.3

Cancel the common factor of $3$ .

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Step 16.2.1.3.1

Cancel the common factor.

$f (\frac{4}{3}) = - 2 - 4 \sin (\frac{π}{2} \cdot \frac{3}{3})$

Step 16.2.1.3.2

Rewrite the expression.

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

Step 16.2.1.4

Multiply $\frac{π}{2}$ by $1$ .

$f (\frac{4}{3}) = - 2 - 4 \sin (\frac{π}{2})$

Step 16.2.1.5

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

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

Step 16.2.1.6

Multiply $- 4$ by $1$ .

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

Step 16.2.2

Subtract $4$ from $- 2$ .

$f (\frac{4}{3}) = - 6$

Step 16.2.3

The final answer is $- 6$ .

$y = - 6$

Step 17

Evaluate the second derivative at $x = \frac{10}{3}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$π^{2} \sin (\frac{π (\frac{10}{3})}{2} - \frac{π}{6})$

Step 18

Evaluate the second derivative.

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Step 18.1

Simplify each term.

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Step 18.1.1

Combine $π$ and $\frac{10}{3}$ .

$π^{2} \sin (\frac{\frac{π \cdot 10}{3}}{2} - \frac{π}{6})$

Step 18.1.2

Move $10$ to the left of $π$ .

$π^{2} \sin (\frac{\frac{10 π}{3}}{2} - \frac{π}{6})$

Step 18.1.3

Multiply the numerator by the reciprocal of the denominator.

$π^{2} \sin (\frac{10 π}{3} \cdot \frac{1}{2} - \frac{π}{6})$

Step 18.1.4

Cancel the common factor of $2$ .

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Step 18.1.4.1

Factor $2$ out of $10 π$ .

$π^{2} \sin (\frac{2 (5 π)}{3} \cdot \frac{1}{2} - \frac{π}{6})$

Step 18.1.4.2

Cancel the common factor.

$π^{2} \sin (\frac{2 (5 π)}{3} \cdot \frac{1}{2} - \frac{π}{6})$

Step 18.1.4.3

Rewrite the expression.

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

Step 18.2

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

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

Step 18.3

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

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Step 18.3.1

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

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

Step 18.3.2

Multiply $3$ by $2$ .

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

Step 18.4

Combine the numerators over the common denominator.

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

Step 18.5

Simplify the numerator.

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Step 18.5.1

Multiply $2$ by $5$ .

$π^{2} \sin (\frac{10 π - π}{6})$

Step 18.5.2

Subtract $π$ from $10 π$ .

$π^{2} \sin (\frac{9 π}{6})$

Step 18.6

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

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Step 18.6.1

Factor $3$ out of $9 π$ .

$π^{2} \sin (\frac{3 (3 π)}{6})$

Step 18.6.2

Cancel the common factors.

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Step 18.6.2.1

Factor $3$ out of $6$ .

$π^{2} \sin (\frac{3 (3 π)}{3 (2)})$

Step 18.6.2.2

Cancel the common factor.

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

Step 18.6.2.3

Rewrite the expression.

$π^{2} \sin (\frac{3 π}{2})$

Step 18.7

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.

$π^{2} (- \sin (\frac{π}{2}))$

Step 18.8

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

$π^{2} (- 1 \cdot 1)$

Step 18.9

Multiply $- 1$ by $1$ .

$π^{2} \cdot - 1$

Step 18.10

Move $- 1$ to the left of $π^{2}$ .

$- 1 \cdot π^{2}$

Step 18.11

Rewrite $- 1 π^{2}$ as $- π^{2}$ .

$- π^{2}$

Step 19

$x = \frac{10}{3}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = \frac{10}{3}$ is a local maximum

Step 20

Find the y-value when $x = \frac{10}{3}$ .

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Step 20.1

Replace the variable $x$ with $\frac{10}{3}$ in the expression.

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

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

Combine the numerators over the common denominator.

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

Step 20.2.1.2

Subtract $1$ from $10$ .

$f (\frac{10}{3}) = - 2 - 4 \sin (\frac{π}{2} \cdot \frac{9}{3})$

Step 20.2.1.3

Divide $9$ by $3$ .

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

Step 20.2.1.4

Combine $\frac{π}{2}$ and $3$ .

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

Step 20.2.1.5

Move $3$ to the left of $π$ .

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

Step 20.2.1.6

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.

$f (\frac{10}{3}) = - 2 - 4 (- \sin (\frac{π}{2}))$

Step 20.2.1.7

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

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

Step 20.2.1.8

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

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Step 20.2.1.8.1

Multiply $- 1$ by $1$ .

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

Step 20.2.1.8.2

Multiply $- 4$ by $- 1$ .

$f (\frac{10}{3}) = - 2 + 4$

Step 20.2.2

Add $- 2$ and $4$ .

$f (\frac{10}{3}) = 2$

Step 20.2.3

The final answer is $2$ .

$y = 2$

Step 21

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

$(\frac{4}{3}, - 6)$ is a local minima

$(\frac{10}{3}, 2)$ is a local maxima

Step 22