Calculus Examples

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

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

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

Step 1.1.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

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

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Step 1.2.3.1

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

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

Step 1.2.3.2

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

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

Step 1.2.3.3

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

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

Step 1.2.4

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

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

Step 1.2.5

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

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

Step 1.2.6

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

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

Step 1.2.7

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

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

Step 1.2.8

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

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

Step 1.2.9

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

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

Step 1.2.10

Multiply $2$ by $- 3$ .

$0 + \frac{- 6 \cos (\frac{2 x}{3})}{3}$

Step 1.2.11

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

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Step 1.2.11.1

Factor $3$ out of $- 6 \cos (\frac{2 x}{3})$ .

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

Step 1.2.11.2

Cancel the common factors.

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Step 1.2.11.2.1

Factor $3$ out of $3$ .

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

Step 1.2.11.2.2

Cancel the common factor.

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

Step 1.2.11.2.3

Rewrite the expression.

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

Step 1.2.11.2.4

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

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

Step 1.3

Subtract $2 \cos (\frac{2 x}{3})$ from $0$ .

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

Step 2

Find the second derivative of the function.

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Step 2.1

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

$f'' (x) = - 2 \frac{d}{d x} \cos (\frac{2 x}{3})$

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 $f (x) = \cos (x)$ and $g (x) = \frac{2 x}{3}$ .

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Step 2.2.1

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

$f'' (x) = - 2 (\frac{d}{d u} (\cos (u)) \frac{d}{d x} (\frac{2 x}{3}))$

Step 2.2.2

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

$f'' (x) = - 2 (- \sin (u) \frac{d}{d x} \frac{2 x}{3})$

Step 2.2.3

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

$f'' (x) = - 2 (- \sin (\frac{2 x}{3}) \frac{d}{d x} \frac{2 x}{3})$

Step 2.3

Differentiate.

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Step 2.3.1

Multiply $- 1$ by $- 2$ .

$f'' (x) = 2 (\sin (\frac{2 x}{3}) \frac{d}{d x} (\frac{2 x}{3}))$

Step 2.3.2

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

$f'' (x) = 2 \sin (\frac{2 x}{3}) (\frac{2}{3} \cdot \frac{d}{d x} (x))$

Step 2.3.3

Combine fractions.

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Step 2.3.3.1

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

$f'' (x) = \frac{2 \cdot 2}{3} \cdot \sin (\frac{2 x}{3}) \frac{d}{d x} (x)$

Step 2.3.3.2

Multiply $2$ by $2$ .

$f'' (x) = \frac{4}{3} \cdot \sin (\frac{2 x}{3}) \frac{d}{d x} (x)$

Step 2.3.3.3

Combine $\frac{4}{3}$ and $\sin (\frac{2 x}{3})$ .

$f'' (x) = \frac{4 \sin (\frac{2 x}{3})}{3} \cdot \frac{d}{d x} (x)$

Step 2.3.4

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

$f'' (x) = \frac{4 \sin (\frac{2 x}{3})}{3} \cdot 1$

Step 2.3.5

Multiply $\frac{4 \sin (\frac{2 x}{3})}{3}$ by $1$ .

$f'' (x) = \frac{4 \sin (\frac{2 x}{3})}{3}$

Step 3

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

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

Step 4

Divide each term in $- 2 \cos (\frac{2 x}{3}) = 0$ by $- 2$ and simplify.

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Step 4.1

Divide each term in $- 2 \cos (\frac{2 x}{3}) = 0$ by $- 2$ .

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

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.

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

Step 4.2.1.2

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

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

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $0$ by $- 2$ .

$\cos (\frac{2 x}{3}) = 0$

Step 5

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

$\frac{2 x}{3} = \arccos (0)$

Step 6

Simplify the right side.

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Step 6.1

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

$\frac{2 x}{3} = \frac{π}{2}$

Step 7

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

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

Step 8

Simplify both sides of the equation.

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Step 8.1

Simplify the left side.

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Step 8.1.1

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

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Step 8.1.1.1

Cancel the common factor of $3$ .

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Step 8.1.1.1.1

Cancel the common factor.

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

Step 8.1.1.1.2

Rewrite the expression.

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

Step 8.1.1.2

Cancel the common factor of $2$ .

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Step 8.1.1.2.1

Factor $2$ out of $2 x$ .

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

Step 8.1.1.2.2

Cancel the common factor.

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

Step 8.1.1.2.3

Rewrite the expression.

$x = \frac{3}{2} \cdot \frac{π}{2}$

Step 8.2

Simplify the right side.

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Step 8.2.1

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

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Step 8.2.1.1

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

$x = \frac{3 π}{2 \cdot 2}$

Step 8.2.1.2

Multiply $2$ by $2$ .

$x = \frac{3 π}{4}$

Step 9

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.

$\frac{2 x}{3} = 2 π - \frac{π}{2}$

Step 10

Solve for $x$ .

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Step 10.1

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

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

Step 10.2

Simplify both sides of the equation.

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Step 10.2.1

Simplify the left side.

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Step 10.2.1.1

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

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Step 10.2.1.1.1

Cancel the common factor of $3$ .

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Step 10.2.1.1.1.1

Cancel the common factor.

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

Step 10.2.1.1.1.2

Rewrite the expression.

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

Step 10.2.1.1.2

Cancel the common factor of $2$ .

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Step 10.2.1.1.2.1

Factor $2$ out of $2 x$ .

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

Step 10.2.1.1.2.2

Cancel the common factor.

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

Step 10.2.1.1.2.3

Rewrite the expression.

$x = \frac{3}{2} (2 π - \frac{π}{2})$

Step 10.2.2

Simplify the right side.

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Step 10.2.2.1

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

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Step 10.2.2.1.1

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

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

Step 10.2.2.1.2

Combine fractions.

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Step 10.2.2.1.2.1

Combine $2 π$ and $\frac{2}{2}$ .

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

Step 10.2.2.1.2.2

Combine the numerators over the common denominator.

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

Step 10.2.2.1.3

Simplify the numerator.

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Step 10.2.2.1.3.1

Multiply $2$ by $2$ .

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

Step 10.2.2.1.3.2

Subtract $π$ from $4 π$ .

$x = \frac{3}{2} \cdot \frac{3 π}{2}$

Step 10.2.2.1.4

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

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Step 10.2.2.1.4.1

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

$x = \frac{3 (3 π)}{2 \cdot 2}$

Step 10.2.2.1.4.2

Multiply $3$ by $3$ .

$x = \frac{9 π}{2 \cdot 2}$

Step 10.2.2.1.4.3

Multiply $2$ by $2$ .

$x = \frac{9 π}{4}$

Step 11

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

$x = \frac{3 π}{4}, \frac{9 π}{4}$

Step 12

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

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

Step 13

Evaluate the second derivative.

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Step 13.1

Combine $2$ and $\frac{3 π}{4}$ .

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

Step 13.2

Multiply $2$ by $3$ .

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

Step 13.3

Reduce the expression $\frac{6 π}{4}$ by cancelling the common factors.

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Step 13.3.1

Factor $2$ out of $6 π$ .

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

Step 13.3.2

Factor $2$ out of $4$ .

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

Step 13.3.3

Cancel the common factor.

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

Step 13.3.4

Rewrite the expression.

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

Step 13.4

Simplify the numerator.

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Step 13.4.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 13.4.2

Cancel the common factor of $3$ .

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Step 13.4.2.1

Factor $3$ out of $3 π$ .

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

Step 13.4.2.2

Cancel the common factor.

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

Step 13.4.2.3

Rewrite the expression.

$\frac{4 \sin (\frac{π}{2})}{3}$

Step 13.4.3

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

$\frac{4 \cdot 1}{3}$

Step 13.5

Multiply $4$ by $1$ .

$\frac{4}{3}$

Step 14

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

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

Step 15

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

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Step 15.1

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

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

Step 15.2

Simplify the result.

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Step 15.2.1

Simplify each term.

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Step 15.2.1.1

Cancel the common factor of $2$ .

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Step 15.2.1.1.1

Factor $2$ out of $4$ .

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

Step 15.2.1.1.2

Cancel the common factor.

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

Step 15.2.1.1.3

Rewrite the expression.

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

Step 15.2.1.2

Cancel the common factor of $3$ .

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Step 15.2.1.2.1

Factor $3$ out of $3 π$ .

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

Step 15.2.1.2.2

Cancel the common factor.

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

Step 15.2.1.2.3

Rewrite the expression.

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

Step 15.2.1.3

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

$f (\frac{3 π}{4}) = 1 - 3 \cdot 1$

Step 15.2.1.4

Multiply $- 3$ by $1$ .

$f (\frac{3 π}{4}) = 1 - 3$

Step 15.2.2

Subtract $3$ from $1$ .

$f (\frac{3 π}{4}) = - 2$

Step 15.2.3

The final answer is $- 2$ .

$y = - 2$

Step 16

Evaluate the second derivative at $x = \frac{9 π}{4}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{4 \sin (\frac{2 (\frac{9 π}{4})}{3})}{3}$

Step 17

Evaluate the second derivative.

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Step 17.1

Combine $2$ and $\frac{9 π}{4}$ .

$\frac{4 \sin (\frac{\frac{2 (9 π)}{4}}{3})}{3}$

Step 17.2

Multiply $2$ by $9$ .

$\frac{4 \sin (\frac{\frac{18 π}{4}}{3})}{3}$

Step 17.3

Reduce the expression $\frac{18 π}{4}$ by cancelling the common factors.

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Step 17.3.1

Factor $2$ out of $18 π$ .

$\frac{4 \sin (\frac{\frac{2 (9 π)}{4}}{3})}{3}$

Step 17.3.2

Factor $2$ out of $4$ .

$\frac{4 \sin (\frac{\frac{2 (9 π)}{2 (2)}}{3})}{3}$

Step 17.3.3

Cancel the common factor.

$\frac{4 \sin (\frac{\frac{2 (9 π)}{2 \cdot 2}}{3})}{3}$

Step 17.3.4

Rewrite the expression.

$\frac{4 \sin (\frac{\frac{9 π}{2}}{3})}{3}$

Step 17.4

Simplify the numerator.

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Step 17.4.1

Multiply the numerator by the reciprocal of the denominator.

$\frac{4 \sin (\frac{9 π}{2} \cdot \frac{1}{3})}{3}$

Step 17.4.2

Cancel the common factor of $3$ .

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Step 17.4.2.1

Factor $3$ out of $9 π$ .

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

Step 17.4.2.2

Cancel the common factor.

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

Step 17.4.2.3

Rewrite the expression.

$\frac{4 \sin (\frac{3 π}{2})}{3}$

Step 17.4.3

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.

$\frac{4 (- \sin (\frac{π}{2}))}{3}$

Step 17.4.4

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

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

Step 17.4.5

Multiply $- 1$ by $1$ .

$\frac{4 \cdot - 1}{3}$

Step 17.5

Simplify the expression.

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Step 17.5.1

Multiply $4$ by $- 1$ .

$\frac{- 4}{3}$

Step 17.5.2

Move the negative in front of the fraction.

$- \frac{4}{3}$

Step 18

$x = \frac{9 π}{4}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = \frac{9 π}{4}$ is a local maximum

Step 19

Find the y-value when $x = \frac{9 π}{4}$ .

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Step 19.1

Replace the variable $x$ with $\frac{9 π}{4}$ in the expression.

$f (\frac{9 π}{4}) = 1 - 3 \sin (\frac{2}{3} \cdot (\frac{9 π}{4}))$

Step 19.2

Simplify the result.

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Step 19.2.1

Simplify each term.

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Step 19.2.1.1

Cancel the common factor of $2$ .

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Step 19.2.1.1.1

Factor $2$ out of $4$ .

$f (\frac{9 π}{4}) = 1 - 3 \sin (\frac{2}{3} \cdot \frac{9 π}{2 (2)})$

Step 19.2.1.1.2

Cancel the common factor.

$f (\frac{9 π}{4}) = 1 - 3 \sin (\frac{2}{3} \cdot \frac{9 π}{2 \cdot 2})$

Step 19.2.1.1.3

Rewrite the expression.

$f (\frac{9 π}{4}) = 1 - 3 \sin (\frac{1}{3} \cdot \frac{9 π}{2})$

Step 19.2.1.2

Cancel the common factor of $3$ .

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Step 19.2.1.2.1

Factor $3$ out of $9 π$ .

$f (\frac{9 π}{4}) = 1 - 3 \sin (\frac{1}{3} \cdot \frac{3 (3 π)}{2})$

Step 19.2.1.2.2

Cancel the common factor.

$f (\frac{9 π}{4}) = 1 - 3 \sin (\frac{1}{3} \cdot \frac{3 (3 π)}{2})$

Step 19.2.1.2.3

Rewrite the expression.

$f (\frac{9 π}{4}) = 1 - 3 \sin (\frac{3 π}{2})$

Step 19.2.1.3

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.

$f (\frac{9 π}{4}) = 1 - 3 (- \sin (\frac{π}{2}))$

Step 19.2.1.4

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

$f (\frac{9 π}{4}) = 1 - 3 (- 1 \cdot 1)$

Step 19.2.1.5

Multiply $- 3 (- 1 \cdot 1)$ .

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Step 19.2.1.5.1

Multiply $- 1$ by $1$ .

$f (\frac{9 π}{4}) = 1 - 3 \cdot - 1$

Step 19.2.1.5.2

Multiply $- 3$ by $- 1$ .

$f (\frac{9 π}{4}) = 1 + 3$

Step 19.2.2

Add $1$ and $3$ .

$f (\frac{9 π}{4}) = 4$

Step 19.2.3

The final answer is $4$ .

$y = 4$

Step 20

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

$(\frac{3 π}{4}, - 2)$ is a local minima

$(\frac{9 π}{4}, 4)$ is a local maxima

Step 21