Algebra Examples

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

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

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

Step 1.1.2

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

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

Step 1.2

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

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Step 1.2.1

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

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

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

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Step 1.2.2.1

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

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

Step 1.2.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

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

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

Step 1.2.5

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

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

Step 1.2.6

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

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

Step 1.2.7

Add $1$ and $0$ .

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

Step 1.2.8

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

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

Step 1.2.9

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

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

Step 1.2.10

Move $2$ to the left of $\cos (\frac{2}{3} (x - 1))$ .

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

Step 1.2.11

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

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

Step 1.2.12

Multiply $2$ by $- 4$ .

$0 + \frac{- 8 \cos (\frac{2}{3} (x - 1))}{3}$

Step 1.2.13

Move the negative in front of the fraction.

$0 - \frac{8 \cos (\frac{2}{3} (x - 1))}{3}$

Step 1.3

Simplify.

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Step 1.3.1

Apply the distributive property.

$0 - \frac{8 \cos (\frac{2}{3} x + \frac{2}{3} \cdot - 1)}{3}$

Step 1.3.2

Combine terms.

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Step 1.3.2.1

Combine $\frac{2}{3}$ and $- 1$ .

$0 - \frac{8 \cos (\frac{2}{3} x + \frac{2 \cdot - 1}{3})}{3}$

Step 1.3.2.2

Multiply $2$ by $- 1$ .

$0 - \frac{8 \cos (\frac{2}{3} x + \frac{- 2}{3})}{3}$

Step 1.3.2.3

Move the negative in front of the fraction.

$0 - \frac{8 \cos (\frac{2}{3} x - \frac{2}{3})}{3}$

Step 1.3.2.4

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

$0 - \frac{8 \cos (\frac{2 x}{3} - \frac{2}{3})}{3}$

Step 1.3.2.5

Subtract $\frac{8 \cos (\frac{2 x}{3} - \frac{2}{3})}{3}$ from $0$ .

$- \frac{8 \cos (\frac{2 x}{3} - \frac{2}{3})}{3}$

Step 2

Find the second derivative of the function.

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Step 2.1

Since $- \frac{8}{3}$ is constant with respect to $x$ , the derivative of $- \frac{8 \cos (\frac{2 x}{3} - \frac{2}{3})}{3}$ with respect to $x$ is $- \frac{8}{3} \frac{d}{d x} [\cos (\frac{2 x}{3} - \frac{2}{3})]$ .

$f'' (x) = - \frac{8}{3} \cdot \frac{d}{d x} \cos (\frac{2 x}{3} - \frac{2}{3})$

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

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Step 2.2.1

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

$f'' (x) = - \frac{8}{3} \cdot (\frac{d}{d u} (\cos (u)) \frac{d}{d x} (\frac{2 x}{3} - \frac{2}{3}))$

Step 2.2.2

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

$f'' (x) = - \frac{8}{3} \cdot (- \sin (u) \frac{d}{d x} (\frac{2 x}{3} - \frac{2}{3}))$

Step 2.2.3

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

$f'' (x) = - \frac{8}{3} \cdot (- \sin (\frac{2 x}{3} - \frac{2}{3}) \frac{d}{d x} (\frac{2 x}{3} - \frac{2}{3}))$

Step 2.3

Differentiate.

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Step 2.3.1

Multiply $- 1$ by $- 1$ .

$f'' (x) = 1 (\frac{8}{3}) \cdot (\sin (\frac{2 x}{3} - \frac{2}{3}) \frac{d}{d x} (\frac{2 x}{3} - \frac{2}{3}))$

Step 2.3.2

Combine fractions.

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Step 2.3.2.1

Multiply $\frac{8}{3}$ by $1$ .

$f'' (x) = \frac{8}{3} \cdot (\sin (\frac{2 x}{3} - \frac{2}{3}) \frac{d}{d x} (\frac{2 x}{3} - \frac{2}{3}))$

Step 2.3.2.2

Combine $\sin (\frac{2 x}{3} - \frac{2}{3})$ and $\frac{8}{3}$ .

$f'' (x) = \frac{\sin (\frac{2 x}{3} - \frac{2}{3}) \cdot 8}{3} \cdot \frac{d}{d x} (\frac{2 x}{3} - \frac{2}{3})$

Step 2.3.2.3

Move $8$ to the left of $\sin (\frac{2 x}{3} - \frac{2}{3})$ .

$f'' (x) = \frac{8 \cdot \sin (\frac{2 x}{3} - \frac{2}{3})}{3} \cdot \frac{d}{d x} (\frac{2 x}{3} - \frac{2}{3})$

Step 2.3.3

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

$f'' (x) = \frac{8 \sin (\frac{2 x}{3} - \frac{2}{3})}{3} \cdot (\frac{d}{d x} (\frac{2 x}{3}) + \frac{d}{d x} (- \frac{2}{3}))$

Step 2.3.4

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

$f'' (x) = \frac{8 \sin (\frac{2 x}{3} - \frac{2}{3})}{3} \cdot (\frac{2}{3} \cdot \frac{d}{d x} (x) + \frac{d}{d x} (- \frac{2}{3}))$

Step 2.3.5

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

$f'' (x) = \frac{8 \sin (\frac{2 x}{3} - \frac{2}{3})}{3} \cdot (\frac{2}{3} \cdot 1 + \frac{d}{d x} (- \frac{2}{3}))$

Step 2.3.6

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

$f'' (x) = \frac{8 \sin (\frac{2 x}{3} - \frac{2}{3})}{3} \cdot (\frac{2}{3} + \frac{d}{d x} (- \frac{2}{3}))$

Step 2.3.7

Since $- \frac{2}{3}$ is constant with respect to $x$ , the derivative of $- \frac{2}{3}$ with respect to $x$ is $0$ .

$f'' (x) = \frac{8 \sin (\frac{2 x}{3} - \frac{2}{3})}{3} \cdot (\frac{2}{3} + 0)$

Step 2.3.8

Combine fractions.

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Step 2.3.8.1

Add $\frac{2}{3}$ and $0$ .

$f'' (x) = \frac{8 \sin (\frac{2 x}{3} - \frac{2}{3})}{3} \cdot \frac{2}{3}$

Step 2.3.8.2

Multiply $\frac{8 \sin (\frac{2 x}{3} - \frac{2}{3})}{3}$ by $\frac{2}{3}$ .

$f'' (x) = \frac{8 \sin (\frac{2 x}{3} - \frac{2}{3}) \cdot 2}{3 \cdot 3}$

Step 2.3.8.3

Multiply.

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Step 2.3.8.3.1

Multiply $2$ by $8$ .

$f'' (x) = \frac{16 \sin (\frac{2 x}{3} - \frac{2}{3})}{3 \cdot 3}$

Step 2.3.8.3.2

Multiply $3$ by $3$ .

$f'' (x) = \frac{16 \sin (\frac{2 x}{3} - \frac{2}{3})}{9}$

Step 3

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

$- \frac{8 \cos (\frac{2 x}{3} - \frac{2}{3})}{3} = 0$

Step 4

Set the numerator equal to zero.

$8 \cos (\frac{2 x}{3} - \frac{2}{3}) = 0$

Step 5

Solve the equation for $x$ .

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Step 5.1

Divide each term in $8 \cos (\frac{2 x}{3} - \frac{2}{3}) = 0$ by $8$ and simplify.

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Step 5.1.1

Divide each term in $8 \cos (\frac{2 x}{3} - \frac{2}{3}) = 0$ by $8$ .

$\frac{8 \cos (\frac{2 x}{3} - \frac{2}{3})}{8} = \frac{0}{8}$

Step 5.1.2

Simplify the left side.

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Step 5.1.2.1

Cancel the common factor of $8$ .

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Step 5.1.2.1.1

Cancel the common factor.

$\frac{8 \cos (\frac{2 x}{3} - \frac{2}{3})}{8} = \frac{0}{8}$

Step 5.1.2.1.2

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

$\cos (\frac{2 x}{3} - \frac{2}{3}) = \frac{0}{8}$

Step 5.1.3

Simplify the right side.

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Step 5.1.3.1

Divide $0$ by $8$ .

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

Step 5.2

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

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

Step 5.3

Simplify the right side.

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Step 5.3.1

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

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

Step 5.4

Add $\frac{2}{3}$ to both sides of the equation.

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

Step 5.5

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

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

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

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Step 5.6.1.1.1

Cancel the common factor of $3$ .

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Step 5.6.1.1.1.1

Cancel the common factor.

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

Step 5.6.1.1.1.2

Rewrite the expression.

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

Step 5.6.1.1.2

Cancel the common factor of $2$ .

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Step 5.6.1.1.2.1

Factor $2$ out of $2 x$ .

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

Step 5.6.1.1.2.2

Cancel the common factor.

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

Step 5.6.1.1.2.3

Rewrite the expression.

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

Step 5.6.2

Simplify the right side.

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Step 5.6.2.1

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

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Step 5.6.2.1.1

Apply the distributive property.

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

Step 5.6.2.1.2

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

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Step 5.6.2.1.2.1

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

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

Step 5.6.2.1.2.2

Multiply $2$ by $2$ .

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

Step 5.6.2.1.3

Cancel the common factor of $3$ .

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Step 5.6.2.1.3.1

Cancel the common factor.

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

Step 5.6.2.1.3.2

Rewrite the expression.

$x = \frac{3 π}{4} + \frac{1}{2} \cdot 2$

Step 5.6.2.1.4

Cancel the common factor of $2$ .

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Step 5.6.2.1.4.1

Cancel the common factor.

$x = \frac{3 π}{4} + \frac{1}{2} \cdot 2$

Step 5.6.2.1.4.2

Rewrite the expression.

$x = \frac{3 π}{4} + 1$

Step 5.7

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.

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

Step 5.8

Solve for $x$ .

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Step 5.8.1

Simplify $2 π - \frac{π}{2}$ .

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Step 5.8.1.1

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

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

Step 5.8.1.2

Combine fractions.

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Step 5.8.1.2.1

Combine $2 π$ and $\frac{2}{2}$ .

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

Step 5.8.1.2.2

Combine the numerators over the common denominator.

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

Step 5.8.1.3

Simplify the numerator.

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Step 5.8.1.3.1

Multiply $2$ by $2$ .

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

Step 5.8.1.3.2

Subtract $π$ from $4 π$ .

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

Step 5.8.2

Add $\frac{2}{3}$ to both sides of the equation.

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

Step 5.8.3

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

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

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

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Step 5.8.4.1.1.1

Cancel the common factor of $3$ .

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Step 5.8.4.1.1.1.1

Cancel the common factor.

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

Step 5.8.4.1.1.1.2

Rewrite the expression.

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

Step 5.8.4.1.1.2

Cancel the common factor of $2$ .

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Step 5.8.4.1.1.2.1

Factor $2$ out of $2 x$ .

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

Step 5.8.4.1.1.2.2

Cancel the common factor.

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

Step 5.8.4.1.1.2.3

Rewrite the expression.

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

Step 5.8.4.2

Simplify the right side.

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Step 5.8.4.2.1

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

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Step 5.8.4.2.1.1

Apply the distributive property.

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

Step 5.8.4.2.1.2

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

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Step 5.8.4.2.1.2.1

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

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

Step 5.8.4.2.1.2.2

Multiply $3$ by $3$ .

$x = \frac{9 π}{2 \cdot 2} + \frac{3}{2} \cdot \frac{2}{3}$

Step 5.8.4.2.1.2.3

Multiply $2$ by $2$ .

$x = \frac{9 π}{4} + \frac{3}{2} \cdot \frac{2}{3}$

Step 5.8.4.2.1.3

Cancel the common factor of $3$ .

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Step 5.8.4.2.1.3.1

Cancel the common factor.

$x = \frac{9 π}{4} + \frac{3}{2} \cdot \frac{2}{3}$

Step 5.8.4.2.1.3.2

Rewrite the expression.

$x = \frac{9 π}{4} + \frac{1}{2} \cdot 2$

Step 5.8.4.2.1.4

Cancel the common factor of $2$ .

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Step 5.8.4.2.1.4.1

Cancel the common factor.

$x = \frac{9 π}{4} + \frac{1}{2} \cdot 2$

Step 5.8.4.2.1.4.2

Rewrite the expression.

$x = \frac{9 π}{4} + 1$

Step 5.9

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

$x = \frac{3 π}{4} + 1, \frac{9 π}{4} + 1$

Step 6

Evaluate the second derivative at $x = \frac{3 π}{4} + 1$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{16 \sin (\frac{2 (\frac{3 π}{4} + 1)}{3} - \frac{2}{3})}{9}$

Step 7

Evaluate the second derivative.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Write $1$ as a fraction with a common denominator.

$\frac{16 \sin (\frac{2 (\frac{3 π}{4} + \frac{4}{4})}{3} - \frac{2}{3})}{9}$

Step 7.1.2

Combine the numerators over the common denominator.

$\frac{16 \sin (\frac{2 \frac{3 π + 4}{4}}{3} - \frac{2}{3})}{9}$

Step 7.2

Simplify terms.

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Step 7.2.1

Combine $2$ and $\frac{3 π + 4}{4}$ .

$\frac{16 \sin (\frac{\frac{2 (3 π + 4)}{4}}{3} - \frac{2}{3})}{9}$

Step 7.2.2

Reduce the expression $\frac{2 (3 π + 4)}{4}$ by cancelling the common factors.

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Step 7.2.2.1

Factor $2$ out of $4$ .

$\frac{16 \sin (\frac{\frac{2 (3 π + 4)}{2 \cdot 2}}{3} - \frac{2}{3})}{9}$

Step 7.2.2.2

Cancel the common factor.

$\frac{16 \sin (\frac{\frac{2 (3 π + 4)}{2 \cdot 2}}{3} - \frac{2}{3})}{9}$

Step 7.2.2.3

Rewrite the expression.

$\frac{16 \sin (\frac{\frac{3 π + 4}{2}}{3} - \frac{2}{3})}{9}$

Step 7.3

Simplify the numerator.

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Step 7.3.1

Combine the numerators over the common denominator.

$\frac{16 \sin (\frac{\frac{3 π + 4}{2} - 2}{3})}{9}$

Step 7.3.2

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

$\frac{16 \sin (\frac{\frac{3 π + 4}{2} - 2 \cdot \frac{2}{2}}{3})}{9}$

Step 7.3.3

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

$\frac{16 \sin (\frac{\frac{3 π + 4}{2} + \frac{- 2 \cdot 2}{2}}{3})}{9}$

Step 7.3.4

Combine the numerators over the common denominator.

$\frac{16 \sin (\frac{\frac{3 π + 4 - 2 \cdot 2}{2}}{3})}{9}$

Step 7.3.5

Simplify the numerator.

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Step 7.3.5.1

Multiply $- 2$ by $2$ .

$\frac{16 \sin (\frac{\frac{3 π + 4 - 4}{2}}{3})}{9}$

Step 7.3.5.2

Subtract $4$ from $4$ .

$\frac{16 \sin (\frac{\frac{3 π + 0}{2}}{3})}{9}$

Step 7.3.5.3

Add $3 π$ and $0$ .

$\frac{16 \sin (\frac{\frac{3 π}{2}}{3})}{9}$

Step 7.3.6

Multiply the numerator by the reciprocal of the denominator.

$\frac{16 \sin (\frac{3 π}{2} \cdot \frac{1}{3})}{9}$

Step 7.3.7

Cancel the common factor of $3$ .

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Step 7.3.7.1

Factor $3$ out of $3 π$ .

$\frac{16 \sin (\frac{3 (π)}{2} \cdot \frac{1}{3})}{9}$

Step 7.3.7.2

Cancel the common factor.

$\frac{16 \sin (\frac{3 π}{2} \cdot \frac{1}{3})}{9}$

Step 7.3.7.3

Rewrite the expression.

$\frac{16 \sin (\frac{π}{2})}{9}$

Step 7.3.8

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

$\frac{16 \cdot 1}{9}$

Step 7.4

Multiply $16$ by $1$ .

$\frac{16}{9}$

Step 8

$x = \frac{3 π}{4} + 1$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = \frac{3 π}{4} + 1$ is a local minimum

Step 9

Find the y-value when $x = \frac{3 π}{4} + 1$ .

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Step 9.1

Replace the variable $x$ with $\frac{3 π}{4} + 1$ in the expression.

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

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

Subtract $1$ from $1$ .

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

Step 9.2.1.2

Add $\frac{3 π}{4}$ and $0$ .

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

Step 9.2.1.3

Cancel the common factor of $2$ .

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Step 9.2.1.3.1

Factor $2$ out of $4$ .

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

Step 9.2.1.3.2

Cancel the common factor.

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

Step 9.2.1.3.3

Rewrite the expression.

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

Step 9.2.1.4

Cancel the common factor of $3$ .

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Step 9.2.1.4.1

Factor $3$ out of $3 π$ .

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

Step 9.2.1.4.2

Cancel the common factor.

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

Step 9.2.1.4.3

Rewrite the expression.

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

Step 9.2.1.5

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

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

Step 9.2.1.6

Multiply $- 4$ by $1$ .

$f (\frac{3 π}{4} + 1) = 3 - 4$

Step 9.2.2

Subtract $4$ from $3$ .

$f (\frac{3 π}{4} + 1) = - 1$

Step 9.2.3

The final answer is $- 1$ .

$y = - 1$

Step 10

Evaluate the second derivative at $x = \frac{9 π}{4} + 1$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{16 \sin (\frac{2 (\frac{9 π}{4} + 1)}{3} - \frac{2}{3})}{9}$

Step 11

Evaluate the second derivative.

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Step 11.1

Simplify the numerator.

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Step 11.1.1

Write $1$ as a fraction with a common denominator.

$\frac{16 \sin (\frac{2 (\frac{9 π}{4} + \frac{4}{4})}{3} - \frac{2}{3})}{9}$

Step 11.1.2

Combine the numerators over the common denominator.

$\frac{16 \sin (\frac{2 \frac{9 π + 4}{4}}{3} - \frac{2}{3})}{9}$

Step 11.2

Simplify terms.

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Step 11.2.1

Combine $2$ and $\frac{9 π + 4}{4}$ .

$\frac{16 \sin (\frac{\frac{2 (9 π + 4)}{4}}{3} - \frac{2}{3})}{9}$

Step 11.2.2

Reduce the expression $\frac{2 (9 π + 4)}{4}$ by cancelling the common factors.

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Step 11.2.2.1

Factor $2$ out of $4$ .

$\frac{16 \sin (\frac{\frac{2 (9 π + 4)}{2 \cdot 2}}{3} - \frac{2}{3})}{9}$

Step 11.2.2.2

Cancel the common factor.

$\frac{16 \sin (\frac{\frac{2 (9 π + 4)}{2 \cdot 2}}{3} - \frac{2}{3})}{9}$

Step 11.2.2.3

Rewrite the expression.

$\frac{16 \sin (\frac{\frac{9 π + 4}{2}}{3} - \frac{2}{3})}{9}$

Step 11.3

Simplify the numerator.

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Step 11.3.1

Combine the numerators over the common denominator.

$\frac{16 \sin (\frac{\frac{9 π + 4}{2} - 2}{3})}{9}$

Step 11.3.2

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

$\frac{16 \sin (\frac{\frac{9 π + 4}{2} - 2 \cdot \frac{2}{2}}{3})}{9}$

Step 11.3.3

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

$\frac{16 \sin (\frac{\frac{9 π + 4}{2} + \frac{- 2 \cdot 2}{2}}{3})}{9}$

Step 11.3.4

Combine the numerators over the common denominator.

$\frac{16 \sin (\frac{\frac{9 π + 4 - 2 \cdot 2}{2}}{3})}{9}$

Step 11.3.5

Simplify the numerator.

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Step 11.3.5.1

Multiply $- 2$ by $2$ .

$\frac{16 \sin (\frac{\frac{9 π + 4 - 4}{2}}{3})}{9}$

Step 11.3.5.2

Subtract $4$ from $4$ .

$\frac{16 \sin (\frac{\frac{9 π + 0}{2}}{3})}{9}$

Step 11.3.5.3

Add $9 π$ and $0$ .

$\frac{16 \sin (\frac{\frac{9 π}{2}}{3})}{9}$

Step 11.3.6

Multiply the numerator by the reciprocal of the denominator.

$\frac{16 \sin (\frac{9 π}{2} \cdot \frac{1}{3})}{9}$

Step 11.3.7

Cancel the common factor of $3$ .

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Step 11.3.7.1

Factor $3$ out of $9 π$ .

$\frac{16 \sin (\frac{3 (3 π)}{2} \cdot \frac{1}{3})}{9}$

Step 11.3.7.2

Cancel the common factor.

$\frac{16 \sin (\frac{3 (3 π)}{2} \cdot \frac{1}{3})}{9}$

Step 11.3.7.3

Rewrite the expression.

$\frac{16 \sin (\frac{3 π}{2})}{9}$

Step 11.3.8

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.

$\frac{16 (- \sin (\frac{π}{2}))}{9}$

Step 11.3.9

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

$\frac{16 (- 1 \cdot 1)}{9}$

Step 11.3.10

Multiply $- 1$ by $1$ .

$\frac{16 \cdot - 1}{9}$

Step 11.4

Simplify the expression.

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Step 11.4.1

Multiply $16$ by $- 1$ .

$\frac{- 16}{9}$

Step 11.4.2

Move the negative in front of the fraction.

$- \frac{16}{9}$

Step 12

$x = \frac{9 π}{4} + 1$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = \frac{9 π}{4} + 1$ is a local maximum

Step 13

Find the y-value when $x = \frac{9 π}{4} + 1$ .

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Step 13.1

Replace the variable $x$ with $\frac{9 π}{4} + 1$ in the expression.

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

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

Subtract $1$ from $1$ .

$f (\frac{9 π}{4} + 1) = 3 - 4 \sin (\frac{2}{3} \cdot (\frac{9 π}{4} + 0))$

Step 13.2.1.2

Add $\frac{9 π}{4}$ and $0$ .

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

Step 13.2.1.3

Cancel the common factor of $2$ .

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Step 13.2.1.3.1

Factor $2$ out of $4$ .

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

Step 13.2.1.3.2

Cancel the common factor.

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

Step 13.2.1.3.3

Rewrite the expression.

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

Step 13.2.1.4

Cancel the common factor of $3$ .

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Step 13.2.1.4.1

Factor $3$ out of $9 π$ .

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

Step 13.2.1.4.2

Cancel the common factor.

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

Step 13.2.1.4.3

Rewrite the expression.

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

Step 13.2.1.5

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.

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

Step 13.2.1.6

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

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

Step 13.2.1.7

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

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Step 13.2.1.7.1

Multiply $- 1$ by $1$ .

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

Step 13.2.1.7.2

Multiply $- 4$ by $- 1$ .

$f (\frac{9 π}{4} + 1) = 3 + 4$

Step 13.2.2

Add $3$ and $4$ .

$f (\frac{9 π}{4} + 1) = 7$

Step 13.2.3

The final answer is $7$ .

$y = 7$

Step 14

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

$(\frac{3 π}{4} + 1, - 1)$ is a local minima

$(\frac{9 π}{4} + 1, 7)$ is a local maxima

Step 15