Calculus Examples

$y = - 7 \cos (\frac{2 x}{3} - 4) + 3$

Step 1

Find the first derivative of the function.

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Step 1.1

By the Sum Rule, the derivative of $- 7 \cos (\frac{2 x}{3} - 4) + 3$ with respect to $x$ is $\frac{d}{d x} [- 7 \cos (\frac{2 x}{3} - 4)] + \frac{d}{d x} [3]$ .

$\frac{d}{d x} [- 7 \cos (\frac{2 x}{3} - 4)] + \frac{d}{d x} [3]$

Step 1.2

Evaluate $\frac{d}{d x} [- 7 \cos (\frac{2 x}{3} - 4)]$ .

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Step 1.2.1

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7 \cos (\frac{2 x}{3} - 4)$ with respect to $x$ is $- 7 \frac{d}{d x} [\cos (\frac{2 x}{3} - 4)]$ .

$- 7 \frac{d}{d x} [\cos (\frac{2 x}{3} - 4)] + \frac{d}{d x} [3]$

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 $f (x) = \cos (x)$ and $g (x) = \frac{2 x}{3} - 4$ .

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Step 1.2.2.1

To apply the Chain Rule, set $u$ as $\frac{2 x}{3} - 4$ .

$- 7 (\frac{d}{d u} [\cos (u)] \frac{d}{d x} [\frac{2 x}{3} - 4]) + \frac{d}{d x} [3]$

Step 1.2.2.2

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

$- 7 (- \sin (u) \frac{d}{d x} [\frac{2 x}{3} - 4]) + \frac{d}{d x} [3]$

Step 1.2.2.3

Replace all occurrences of $u$ with $\frac{2 x}{3} - 4$ .

$- 7 (- \sin (\frac{2 x}{3} - 4) \frac{d}{d x} [\frac{2 x}{3} - 4]) + \frac{d}{d x} [3]$

Step 1.2.3

By the Sum Rule, the derivative of $\frac{2 x}{3} - 4$ with respect to $x$ is $\frac{d}{d x} [\frac{2 x}{3}] + \frac{d}{d x} [- 4]$ .

$- 7 (- \sin (\frac{2 x}{3} - 4) (\frac{d}{d x} [\frac{2 x}{3}] + \frac{d}{d x} [- 4])) + \frac{d}{d x} [3]$

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 $\frac{2}{3} \frac{d}{d x} [x]$ .

$- 7 (- \sin (\frac{2 x}{3} - 4) (\frac{2}{3} \frac{d}{d x} [x] + \frac{d}{d x} [- 4])) + \frac{d}{d x} [3]$

Step 1.2.5

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

$- 7 (- \sin (\frac{2 x}{3} - 4) (\frac{2}{3} \cdot 1 + \frac{d}{d x} [- 4])) + \frac{d}{d x} [3]$

Step 1.2.6

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

$- 7 (- \sin (\frac{2 x}{3} - 4) (\frac{2}{3} \cdot 1 + 0)) + \frac{d}{d x} [3]$

Step 1.2.7

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

$- 7 (- \sin (\frac{2 x}{3} - 4) (\frac{2}{3} + 0)) + \frac{d}{d x} [3]$

Step 1.2.8

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

$- 7 (- \sin (\frac{2 x}{3} - 4) \frac{2}{3}) + \frac{d}{d x} [3]$

Step 1.2.9

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

$- 7 (- \frac{2 \sin (\frac{2 x}{3} - 4)}{3}) + \frac{d}{d x} [3]$

Step 1.2.10

Multiply $- 1$ by $- 7$ .

$7 \frac{2 \sin (\frac{2 x}{3} - 4)}{3} + \frac{d}{d x} [3]$

Step 1.2.11

Combine $7$ and $\frac{2 \sin (\frac{2 x}{3} - 4)}{3}$ .

$\frac{7 (2 \sin (\frac{2 x}{3} - 4))}{3} + \frac{d}{d x} [3]$

Step 1.2.12

Multiply $2$ by $7$ .

$\frac{14 \sin (\frac{2 x}{3} - 4)}{3} + \frac{d}{d x} [3]$

Step 1.3

Differentiate using the Constant Rule.

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Step 1.3.1

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

$\frac{14 \sin (\frac{2 x}{3} - 4)}{3} + 0$

Step 1.3.2

Add $\frac{14 \sin (\frac{2 x}{3} - 4)}{3}$ and $0$ .

$\frac{14 \sin (\frac{2 x}{3} - 4)}{3}$

Step 2

Find the second derivative of the function.

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Step 2.1

Since $\frac{14}{3}$ is constant with respect to $x$ , the derivative of $\frac{14 \sin (\frac{2 x}{3} - 4)}{3}$ with respect to $x$ is $\frac{14}{3} \frac{d}{d x} [\sin (\frac{2 x}{3} - 4)]$ .

$f'' (x) = \frac{14}{3} \cdot \frac{d}{d x} (\sin (\frac{2 x}{3} - 4))$

Step 2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = \frac{2 x}{3} - 4$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $\frac{2 x}{3} - 4$ .

$f'' (x) = \frac{14}{3} \cdot (\frac{d}{d u} (\sin (u)) \frac{d}{d x} (\frac{2 x}{3} - 4))$

Step 2.2.2

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

$f'' (x) = \frac{14}{3} \cdot (\cos (u) \frac{d}{d x} (\frac{2 x}{3} - 4))$

Step 2.2.3

Replace all occurrences of $u$ with $\frac{2 x}{3} - 4$ .

$f'' (x) = \frac{14}{3} \cdot (\cos (\frac{2 x}{3} - 4) \frac{d}{d x} (\frac{2 x}{3} - 4))$

Step 2.3

Differentiate.

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Step 2.3.1

Combine $\cos (\frac{2 x}{3} - 4)$ and $\frac{14}{3}$ .

$f'' (x) = \frac{\cos (\frac{2 x}{3} - 4) \cdot 14}{3} \cdot \frac{d}{d x} (\frac{2 x}{3} - 4)$

Step 2.3.2

Move $14$ to the left of $\cos (\frac{2 x}{3} - 4)$ .

$f'' (x) = \frac{14 \cdot \cos (\frac{2 x}{3} - 4)}{3} \cdot \frac{d}{d x} (\frac{2 x}{3} - 4)$

Step 2.3.3

By the Sum Rule, the derivative of $\frac{2 x}{3} - 4$ with respect to $x$ is $\frac{d}{d x} [\frac{2 x}{3}] + \frac{d}{d x} [- 4]$ .

$f'' (x) = \frac{14 \cos (\frac{2 x}{3} - 4)}{3} \cdot (\frac{d}{d x} (\frac{2 x}{3}) + \frac{d}{d x} (- 4))$

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 $\frac{2}{3} \frac{d}{d x} [x]$ .

$f'' (x) = \frac{14 \cos (\frac{2 x}{3} - 4)}{3} \cdot (\frac{2}{3} \cdot \frac{d}{d x} (x) + \frac{d}{d x} (- 4))$

Step 2.3.5

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

$f'' (x) = \frac{14 \cos (\frac{2 x}{3} - 4)}{3} \cdot (\frac{2}{3} \cdot 1 + \frac{d}{d x} (- 4))$

Step 2.3.6

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

$f'' (x) = \frac{14 \cos (\frac{2 x}{3} - 4)}{3} \cdot (\frac{2}{3} + \frac{d}{d x} (- 4))$

Step 2.3.7

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

$f'' (x) = \frac{14 \cos (\frac{2 x}{3} - 4)}{3} \cdot (\frac{2}{3} + 0)$

Step 2.3.8

Combine fractions.

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Step 2.3.8.1

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

$f'' (x) = \frac{14 \cos (\frac{2 x}{3} - 4)}{3} \cdot \frac{2}{3}$

Step 2.3.8.2

Multiply $\frac{14 \cos (\frac{2 x}{3} - 4)}{3}$ by $\frac{2}{3}$ .

$f'' (x) = \frac{14 \cos (\frac{2 x}{3} - 4) \cdot 2}{3 \cdot 3}$

Step 2.3.8.3

Multiply.

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Step 2.3.8.3.1

Multiply $2$ by $14$ .

$f'' (x) = \frac{28 \cos (\frac{2 x}{3} - 4)}{3 \cdot 3}$

Step 2.3.8.3.2

Multiply $3$ by $3$ .

$f'' (x) = \frac{28 \cos (\frac{2 x}{3} - 4)}{9}$

Step 3

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

$\frac{14 \sin (\frac{2 x}{3} - 4)}{3} = 0$

Step 4

Set the numerator equal to zero.

$14 \sin (\frac{2 x}{3} - 4) = 0$

Step 5

Solve the equation for $x$ .

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Step 5.1

Divide each term in $14 \sin (\frac{2 x}{3} - 4) = 0$ by $14$ and simplify.

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Step 5.1.1

Divide each term in $14 \sin (\frac{2 x}{3} - 4) = 0$ by $14$ .

$\frac{14 \sin (\frac{2 x}{3} - 4)}{14} = \frac{0}{14}$

Step 5.1.2

Simplify the left side.

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Step 5.1.2.1

Cancel the common factor of $14$ .

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Step 5.1.2.1.1

Cancel the common factor.

$\frac{14 \sin (\frac{2 x}{3} - 4)}{14} = \frac{0}{14}$

Step 5.1.2.1.2

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

$\sin (\frac{2 x}{3} - 4) = \frac{0}{14}$

Step 5.1.3

Simplify the right side.

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Step 5.1.3.1

Divide $0$ by $14$ .

$\sin (\frac{2 x}{3} - 4) = 0$

Step 5.2

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

$\frac{2 x}{3} - 4 = \arcsin (0)$

Step 5.3

Simplify the right side.

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Step 5.3.1

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

$\frac{2 x}{3} - 4 = 0$

Step 5.4

Add $4$ to both sides of the equation.

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

Step 5.5

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

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

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.

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

Step 5.6.1.1.1.2

Rewrite the expression.

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

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 $2 x$ .

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

Step 5.6.1.1.2.2

Cancel the common factor.

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

Step 5.6.1.1.2.3

Rewrite the expression.

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

Step 5.6.2

Simplify the right side.

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Step 5.6.2.1

Simplify $\frac{3}{2} \cdot 4$ .

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Step 5.6.2.1.1

Cancel the common factor of $2$ .

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Step 5.6.2.1.1.1

Factor $2$ out of $4$ .

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

Step 5.6.2.1.1.2

Cancel the common factor.

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

Step 5.6.2.1.1.3

Rewrite the expression.

$x = 3 \cdot 2$

Step 5.6.2.1.2

Multiply $3$ by $2$ .

$x = 6$

Step 5.7

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

$\frac{2 x}{3} - 4 = π - 0$

Step 5.8

Solve for $x$ .

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Step 5.8.1

Subtract $0$ from $π$ .

$\frac{2 x}{3} - 4 = π$

Step 5.8.2

Add $4$ to both sides of the equation.

$\frac{2 x}{3} = π + 4$

Step 5.8.3

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

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

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.

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

Step 5.8.4.1.1.1.2

Rewrite the expression.

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

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 $2 x$ .

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

Step 5.8.4.1.1.2.2

Cancel the common factor.

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

Step 5.8.4.1.1.2.3

Rewrite the expression.

$x = \frac{3}{2} (π + 4)$

Step 5.8.4.2

Simplify the right side.

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Step 5.8.4.2.1

Simplify $\frac{3}{2} (π + 4)$ .

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Step 5.8.4.2.1.1

Apply the distributive property.

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

Step 5.8.4.2.1.2

Combine $\frac{3}{2}$ and $π$ .

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

Step 5.8.4.2.1.3

Cancel the common factor of $2$ .

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Step 5.8.4.2.1.3.1

Factor $2$ out of $4$ .

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

Step 5.8.4.2.1.3.2

Cancel the common factor.

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

Step 5.8.4.2.1.3.3

Rewrite the expression.

$x = \frac{3 π}{2} + 3 \cdot 2$

Step 5.8.4.2.1.4

Multiply $3$ by $2$ .

$x = \frac{3 π}{2} + 6$

Step 5.9

The solution to the equation $\frac{2 x}{3} - 4 = 0$ .

$x = 6, \frac{3 π}{2} + 6$

Step 6

Evaluate the second derivative at $x = 6$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{28 \cos (\frac{2 (6)}{3} - 4)}{9}$

Step 7

Evaluate the second derivative.

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Step 7.1

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

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Step 7.1.1

Factor $3$ out of $2 (6)$ .

$\frac{28 \cos (\frac{3 (2 \cdot (2))}{3} - 4)}{9}$

Step 7.1.2

Cancel the common factors.

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Step 7.1.2.1

Factor $3$ out of $3$ .

$\frac{28 \cos (\frac{3 (2 \cdot (2))}{3 (1)} - 4)}{9}$

Step 7.1.2.2

Cancel the common factor.

$\frac{28 \cos (\frac{3 (2 \cdot (2))}{3 \cdot 1} - 4)}{9}$

Step 7.1.2.3

Rewrite the expression.

$\frac{28 \cos (\frac{2 \cdot (2)}{1} - 4)}{9}$

Step 7.1.2.4

Divide $2 \cdot (2)$ by $1$ .

$\frac{28 \cos (2 \cdot (2) - 4)}{9}$

Step 7.2

Simplify the numerator.

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Step 7.2.1

Multiply $2$ by $2$ .

$\frac{28 \cos (4 - 4)}{9}$

Step 7.2.2

Subtract $4$ from $4$ .

$\frac{28 \cos (0)}{9}$

Step 7.2.3

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

$\frac{28 \cdot 1}{9}$

Step 7.3

Multiply $28$ by $1$ .

$\frac{28}{9}$

Step 8

$x = 6$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = 6$ is a local minimum

Step 9

Find the y-value when $x = 6$ .

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Step 9.1

Replace the variable $x$ with $6$ in the expression.

$f (6) = - 7 \cos (\frac{2 (6)}{3} - 4) + 3$

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

Multiply $2$ by $6$ .

$f (6) = - 7 \cos (\frac{12}{3} - 4) + 3$

Step 9.2.1.2

Divide $12$ by $3$ .

$f (6) = - 7 \cos (4 - 4) + 3$

Step 9.2.1.3

Subtract $4$ from $4$ .

$f (6) = - 7 \cos (0) + 3$

Step 9.2.1.4

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

$f (6) = - 7 \cdot 1 + 3$

Step 9.2.1.5

Multiply $- 7$ by $1$ .

$f (6) = - 7 + 3$

Step 9.2.2

Add $- 7$ and $3$ .

$f (6) = - 4$

Step 9.2.3

The final answer is $- 4$ .

$y = - 4$

Step 10

Evaluate the second derivative at $x = \frac{3 π}{2} + 6$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{28 \cos (\frac{2 (\frac{3 π}{2} + 6)}{3} - 4)}{9}$

Step 11

Evaluate the second derivative.

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Step 11.1

Cancel the common factor of $\frac{3 π}{2} + 6$ and $3$ .

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Step 11.1.1

Factor $3$ out of $2 (\frac{3 π}{2} + 6)$ .

$\frac{28 \cos (\frac{3 (2 (\frac{π}{2} + 2))}{3} - 4)}{9}$

Step 11.1.2

Cancel the common factors.

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Step 11.1.2.1

Factor $3$ out of $3$ .

$\frac{28 \cos (\frac{3 (2 (\frac{π}{2} + 2))}{3 (1)} - 4)}{9}$

Step 11.1.2.2

Cancel the common factor.

$\frac{28 \cos (\frac{3 (2 (\frac{π}{2} + 2))}{3 \cdot 1} - 4)}{9}$

Step 11.1.2.3

Rewrite the expression.

$\frac{28 \cos (\frac{2 (\frac{π}{2} + 2)}{1} - 4)}{9}$

Step 11.1.2.4

Divide $2 (\frac{π}{2} + 2)$ by $1$ .

$\frac{28 \cos (2 (\frac{π}{2} + 2) - 4)}{9}$

Step 11.2

Simplify the numerator.

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Step 11.2.1

Simplify each term.

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Step 11.2.1.1

Apply the distributive property.

$\frac{28 \cos (2 \frac{π}{2} + 2 \cdot 2 - 4)}{9}$

Step 11.2.1.2

Cancel the common factor of $2$ .

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Step 11.2.1.2.1

Cancel the common factor.

$\frac{28 \cos (2 \frac{π}{2} + 2 \cdot 2 - 4)}{9}$

Step 11.2.1.2.2

Rewrite the expression.

$\frac{28 \cos (π + 2 \cdot 2 - 4)}{9}$

Step 11.2.1.3

Multiply $2$ by $2$ .

$\frac{28 \cos (π + 4 - 4)}{9}$

Step 11.2.2

Subtract $4$ from $4$ .

$\frac{28 \cos (π + 0)}{9}$

Step 11.2.3

Add $π$ and $0$ .

$\frac{28 \cos (π)}{9}$

Step 11.2.4

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

$\frac{28 (- \cos (0))}{9}$

Step 11.2.5

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

$\frac{28 (- 1 \cdot 1)}{9}$

Step 11.2.6

Multiply $- 1$ by $1$ .

$\frac{28 \cdot - 1}{9}$

Step 11.3

Simplify the expression.

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Step 11.3.1

Multiply $28$ by $- 1$ .

$\frac{- 28}{9}$

Step 11.3.2

Move the negative in front of the fraction.

$- \frac{28}{9}$

Step 12

$x = \frac{3 π}{2} + 6$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = \frac{3 π}{2} + 6$ is a local maximum

Step 13

Find the y-value when $x = \frac{3 π}{2} + 6$ .

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Step 13.1

Replace the variable $x$ with $\frac{3 π}{2} + 6$ in the expression.

$f (\frac{3 π}{2} + 6) = - 7 \cos (\frac{2 (\frac{3 π}{2} + 6)}{3} - 4) + 3$

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

Simplify each term.

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Step 13.2.1.1.1

Cancel the common factor of $\frac{3 π}{2} + 6$ and $3$ .

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Step 13.2.1.1.1.1

Factor $3$ out of $2 (\frac{3 π}{2} + 6)$ .

$f (\frac{3 π}{2} + 6) = - 7 \cos (\frac{3 (2 (\frac{π}{2} + 2))}{3} - 4) + 3$

Step 13.2.1.1.1.2

Cancel the common factors.

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Step 13.2.1.1.1.2.1

Factor $3$ out of $3$ .

$f (\frac{3 π}{2} + 6) = - 7 \cos (\frac{3 (2 (\frac{π}{2} + 2))}{3 (1)} - 4) + 3$

Step 13.2.1.1.1.2.2

Cancel the common factor.

$f (\frac{3 π}{2} + 6) = - 7 \cos (\frac{3 (2 (\frac{π}{2} + 2))}{3 \cdot 1} - 4) + 3$

Step 13.2.1.1.1.2.3

Rewrite the expression.

$f (\frac{3 π}{2} + 6) = - 7 \cos (\frac{2 (\frac{π}{2} + 2)}{1} - 4) + 3$

Step 13.2.1.1.1.2.4

Divide $2 (\frac{π}{2} + 2)$ by $1$ .

$f (\frac{3 π}{2} + 6) = - 7 \cos (2 (\frac{π}{2} + 2) - 4) + 3$

Step 13.2.1.1.2

Apply the distributive property.

$f (\frac{3 π}{2} + 6) = - 7 \cos (2 (\frac{π}{2}) + 2 \cdot 2 - 4) + 3$

Step 13.2.1.1.3

Cancel the common factor of $2$ .

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Step 13.2.1.1.3.1

Cancel the common factor.

$f (\frac{3 π}{2} + 6) = - 7 \cos (2 (\frac{π}{2}) + 2 \cdot 2 - 4) + 3$

Step 13.2.1.1.3.2

Rewrite the expression.

$f (\frac{3 π}{2} + 6) = - 7 \cos (π + 2 \cdot 2 - 4) + 3$

Step 13.2.1.1.4

Multiply $2$ by $2$ .

$f (\frac{3 π}{2} + 6) = - 7 \cos (π + 4 - 4) + 3$

Step 13.2.1.2

Subtract $4$ from $4$ .

$f (\frac{3 π}{2} + 6) = - 7 \cos (π + 0) + 3$

Step 13.2.1.3

Add $π$ and $0$ .

$f (\frac{3 π}{2} + 6) = - 7 \cos (π) + 3$

Step 13.2.1.4

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

$f (\frac{3 π}{2} + 6) = - 7 (- \cos (0)) + 3$

Step 13.2.1.5

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

$f (\frac{3 π}{2} + 6) = - 7 (- 1 \cdot 1) + 3$

Step 13.2.1.6

Multiply $- 7 (- 1 \cdot 1)$ .

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Step 13.2.1.6.1

Multiply $- 1$ by $1$ .

$f (\frac{3 π}{2} + 6) = - 7 \cdot - 1 + 3$

Step 13.2.1.6.2

Multiply $- 7$ by $- 1$ .

$f (\frac{3 π}{2} + 6) = 7 + 3$

Step 13.2.2

Add $7$ and $3$ .

$f (\frac{3 π}{2} + 6) = 10$

Step 13.2.3

The final answer is $10$ .

$y = 10$

Step 14

These are the local extrema for $f (x) = - 7 \cos (\frac{2 x}{3} - 4) + 3$ .

$(6, - 4)$ is a local minima

$(\frac{3 π}{2} + 6, 10)$ is a local maxima

Step 15