Calculus Examples

$f (x) = - 5 \sin (\frac{π}{12} x + \frac{π}{3}) + 2$

Step 1

Find the first derivative of the function.

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Step 1.1

By the Sum Rule, the derivative of $- 5 \sin (\frac{π}{12} x + \frac{π}{3}) + 2$ with respect to $x$ is $\frac{d}{d x} [- 5 \sin (\frac{π}{12} x + \frac{π}{3})] + \frac{d}{d x} [2]$ .

$\frac{d}{d x} [- 5 \sin (\frac{π}{12} x + \frac{π}{3})] + \frac{d}{d x} [2]$

Step 1.2

Evaluate $\frac{d}{d x} [- 5 \sin (\frac{π}{12} x + \frac{π}{3})]$ .

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Step 1.2.1

Combine $\frac{π}{12}$ and $x$ .

$\frac{d}{d x} [- 5 \sin (\frac{π x}{12} + \frac{π}{3})] + \frac{d}{d x} [2]$

Step 1.2.2

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 \sin (\frac{π x}{12} + \frac{π}{3})$ with respect to $x$ is $- 5 \frac{d}{d x} [\sin (\frac{π x}{12} + \frac{π}{3})]$ .

$- 5 \frac{d}{d x} [\sin (\frac{π x}{12} + \frac{π}{3})] + \frac{d}{d x} [2]$

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 $g (x) = \frac{π x}{12} + \frac{π}{3}$ .

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Step 1.2.3.1

To apply the Chain Rule, set $u$ as $\frac{π x}{12} + \frac{π}{3}$ .

$- 5 (\frac{d}{d u} [\sin (u)] \frac{d}{d x} [\frac{π x}{12} + \frac{π}{3}]) + \frac{d}{d x} [2]$

Step 1.2.3.2

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

$- 5 (\cos (u) \frac{d}{d x} [\frac{π x}{12} + \frac{π}{3}]) + \frac{d}{d x} [2]$

Step 1.2.3.3

Replace all occurrences of $u$ with $\frac{π x}{12} + \frac{π}{3}$ .

$- 5 (\cos (\frac{π x}{12} + \frac{π}{3}) \frac{d}{d x} [\frac{π x}{12} + \frac{π}{3}]) + \frac{d}{d x} [2]$

Step 1.2.4

By the Sum Rule, the derivative of $\frac{π x}{12} + \frac{π}{3}$ with respect to $x$ is $\frac{d}{d x} [\frac{π x}{12}] + \frac{d}{d x} [\frac{π}{3}]$ .

$- 5 (\cos (\frac{π x}{12} + \frac{π}{3}) (\frac{d}{d x} [\frac{π x}{12}] + \frac{d}{d x} [\frac{π}{3}])) + \frac{d}{d x} [2]$

Step 1.2.5

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

$- 5 (\cos (\frac{π x}{12} + \frac{π}{3}) (\frac{π}{12} \frac{d}{d x} [x] + \frac{d}{d x} [\frac{π}{3}])) + \frac{d}{d x} [2]$

Step 1.2.6

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

$- 5 (\cos (\frac{π x}{12} + \frac{π}{3}) (\frac{π}{12} \cdot 1 + \frac{d}{d x} [\frac{π}{3}])) + \frac{d}{d x} [2]$

Step 1.2.7

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

$- 5 (\cos (\frac{π x}{12} + \frac{π}{3}) (\frac{π}{12} \cdot 1 + 0)) + \frac{d}{d x} [2]$

Step 1.2.8

Multiply $\frac{π}{12}$ by $1$ .

$- 5 (\cos (\frac{π x}{12} + \frac{π}{3}) (\frac{π}{12} + 0)) + \frac{d}{d x} [2]$

Step 1.2.9

Add $\frac{π}{12}$ and $0$ .

$- 5 (\cos (\frac{π x}{12} + \frac{π}{3}) \frac{π}{12}) + \frac{d}{d x} [2]$

Step 1.2.10

Combine $\cos (\frac{π x}{12} + \frac{π}{3})$ and $\frac{π}{12}$ .

$- 5 \frac{\cos (\frac{π x}{12} + \frac{π}{3}) π}{12} + \frac{d}{d x} [2]$

Step 1.2.11

Combine $- 5$ and $\frac{\cos (\frac{π x}{12} + \frac{π}{3}) π}{12}$ .

$\frac{- 5 (\cos (\frac{π x}{12} + \frac{π}{3}) π)}{12} + \frac{d}{d x} [2]$

Step 1.2.12

Move the negative in front of the fraction.

$- \frac{5 \cos (\frac{π x}{12} + \frac{π}{3}) π}{12} + \frac{d}{d x} [2]$

Step 1.3

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

$- \frac{5 \cos (\frac{π x}{12} + \frac{π}{3}) π}{12} + 0$

Step 1.4

Simplify.

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Step 1.4.1

Add $- \frac{5 \cos (\frac{π x}{12} + \frac{π}{3}) π}{12}$ and $0$ .

$- \frac{5 \cos (\frac{π x}{12} + \frac{π}{3}) π}{12}$

Step 1.4.2

Reorder factors in $- \frac{5 \cos (\frac{π x}{12} + \frac{π}{3}) π}{12}$ .

$- \frac{5 π \cos (\frac{π x}{12} + \frac{π}{3})}{12}$

Step 2

Find the second derivative of the function.

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Step 2.1

Since $- \frac{5 π}{12}$ is constant with respect to $x$ , the derivative of $- \frac{5 π \cos (\frac{π x}{12} + \frac{π}{3})}{12}$ with respect to $x$ is $- \frac{5 π}{12} \frac{d}{d x} [\cos (\frac{π x}{12} + \frac{π}{3})]$ .

$f'' (x) = - \frac{5 π}{12} \cdot \frac{d}{d x} \cos (\frac{π x}{12} + \frac{π}{3})$

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 $g (x) = \frac{π x}{12} + \frac{π}{3}$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $\frac{π x}{12} + \frac{π}{3}$ .

$f'' (x) = - \frac{5 π}{12} \cdot (\frac{d}{d u} (\cos (u)) \frac{d}{d x} (\frac{π x}{12} + \frac{π}{3}))$

Step 2.2.2

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

$f'' (x) = - \frac{5 π}{12} \cdot (- \sin (u) \frac{d}{d x} (\frac{π x}{12} + \frac{π}{3}))$

Step 2.2.3

Replace all occurrences of $u$ with $\frac{π x}{12} + \frac{π}{3}$ .

$f'' (x) = - \frac{5 π}{12} \cdot (- \sin (\frac{π x}{12} + \frac{π}{3}) \frac{d}{d x} (\frac{π x}{12} + \frac{π}{3}))$

Step 2.3

Differentiate.

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Step 2.3.1

Multiply $- 1$ by $- 1$ .

$f'' (x) = 1 (\frac{5 π}{12}) \cdot (\sin (\frac{π x}{12} + \frac{π}{3}) \frac{d}{d x} (\frac{π x}{12} + \frac{π}{3}))$

Step 2.3.2

Combine fractions.

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Step 2.3.2.1

Multiply $\frac{5 π}{12}$ by $1$ .

$f'' (x) = \frac{5 π}{12} \cdot (\sin (\frac{π x}{12} + \frac{π}{3}) \frac{d}{d x} (\frac{π x}{12} + \frac{π}{3}))$

Step 2.3.2.2

Combine $\sin (\frac{π x}{12} + \frac{π}{3})$ and $\frac{5 π}{12}$ .

$f'' (x) = \frac{\sin (\frac{π x}{12} + \frac{π}{3}) (5 π)}{12} \cdot \frac{d}{d x} (\frac{π x}{12} + \frac{π}{3})$

Step 2.3.2.3

Move $5$ to the left of $\sin (\frac{π x}{12} + \frac{π}{3})$ .

$f'' (x) = \frac{5 \cdot (\sin (\frac{π x}{12} + \frac{π}{3}) π)}{12} \cdot \frac{d}{d x} (\frac{π x}{12} + \frac{π}{3})$

Step 2.3.3

By the Sum Rule, the derivative of $\frac{π x}{12} + \frac{π}{3}$ with respect to $x$ is $\frac{d}{d x} [\frac{π x}{12}] + \frac{d}{d x} [\frac{π}{3}]$ .

$f'' (x) = \frac{5 \sin (\frac{π x}{12} + \frac{π}{3}) π}{12} \cdot (\frac{d}{d x} (\frac{π x}{12}) + \frac{d}{d x} (\frac{π}{3}))$

Step 2.3.4

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

$f'' (x) = \frac{5 \sin (\frac{π x}{12} + \frac{π}{3}) π}{12} \cdot (\frac{π}{12} \cdot \frac{d}{d x} (x) + \frac{d}{d x} (\frac{π}{3}))$

Step 2.3.5

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

$f'' (x) = \frac{5 \sin (\frac{π x}{12} + \frac{π}{3}) π}{12} \cdot (\frac{π}{12} \cdot 1 + \frac{d}{d x} (\frac{π}{3}))$

Step 2.3.6

Multiply $\frac{π}{12}$ by $1$ .

$f'' (x) = \frac{5 \sin (\frac{π x}{12} + \frac{π}{3}) π}{12} \cdot (\frac{π}{12} + \frac{d}{d x} (\frac{π}{3}))$

Step 2.3.7

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

$f'' (x) = \frac{5 \sin (\frac{π x}{12} + \frac{π}{3}) π}{12} \cdot (\frac{π}{12} + 0)$

Step 2.3.8

Combine fractions.

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Step 2.3.8.1

Add $\frac{π}{12}$ and $0$ .

$f'' (x) = \frac{5 \sin (\frac{π x}{12} + \frac{π}{3}) π}{12} \cdot \frac{π}{12}$

Step 2.3.8.2

Multiply $\frac{5 \sin (\frac{π x}{12} + \frac{π}{3}) π}{12}$ by $\frac{π}{12}$ .

$f'' (x) = \frac{5 \sin (\frac{π x}{12} + \frac{π}{3}) π π}{12 \cdot 12}$

Step 2.4

Raise $π$ to the power of $1$ .

$f'' (x) = \frac{5 \sin (\frac{π x}{12} + \frac{π}{3}) (π π)}{12 \cdot 12}$

Step 2.5

Raise $π$ to the power of $1$ .

$f'' (x) = \frac{5 \sin (\frac{π x}{12} + \frac{π}{3}) (π π)}{12 \cdot 12}$

Step 2.6

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

$f'' (x) = \frac{5 \sin (\frac{π x}{12} + \frac{π}{3}) π^{1 + 1}}{12 \cdot 12}$

Step 2.7

Add $1$ and $1$ .

$f'' (x) = \frac{5 \sin (\frac{π x}{12} + \frac{π}{3}) π^{2}}{12 \cdot 12}$

Step 2.8

Multiply $12$ by $12$ .

$f'' (x) = \frac{5 \sin (\frac{π x}{12} + \frac{π}{3}) π^{2}}{144}$

Step 2.9

Reorder factors in $5 \sin (\frac{π x}{12} + \frac{π}{3}) π^{2}$ .

$f'' (x) = \frac{5 π^{2} \sin (\frac{π x}{12} + \frac{π}{3})}{144}$

Step 3

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

$- \frac{5 π \cos (\frac{π x}{12} + \frac{π}{3})}{12} = 0$

Step 4

Set the numerator equal to zero.

$5 π \cos (\frac{π x}{12} + \frac{π}{3}) = 0$

Step 5

Solve the equation for $x$ .

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Step 5.1

Divide each term in $5 π \cos (\frac{π x}{12} + \frac{π}{3}) = 0$ by $5 π$ and simplify.

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Step 5.1.1

Divide each term in $5 π \cos (\frac{π x}{12} + \frac{π}{3}) = 0$ by $5 π$ .

$\frac{5 π \cos (\frac{π x}{12} + \frac{π}{3})}{5 π} = \frac{0}{5 π}$

Step 5.1.2

Simplify the left side.

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Step 5.1.2.1

Cancel the common factor of $5$ .

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Step 5.1.2.1.1

Cancel the common factor.

$\frac{5 π \cos (\frac{π x}{12} + \frac{π}{3})}{5 π} = \frac{0}{5 π}$

Step 5.1.2.1.2

Rewrite the expression.

$\frac{π \cos (\frac{π x}{12} + \frac{π}{3})}{π} = \frac{0}{5 π}$

Step 5.1.2.2

Cancel the common factor of $π$ .

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Step 5.1.2.2.1

Cancel the common factor.

$\frac{π \cos (\frac{π x}{12} + \frac{π}{3})}{π} = \frac{0}{5 π}$

Step 5.1.2.2.2

Divide $\cos (\frac{π x}{12} + \frac{π}{3})$ by $1$ .

$\cos (\frac{π x}{12} + \frac{π}{3}) = \frac{0}{5 π}$

Step 5.1.3

Simplify the right side.

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Step 5.1.3.1

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

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Step 5.1.3.1.1

Factor $5$ out of $0$ .

$\cos (\frac{π x}{12} + \frac{π}{3}) = \frac{5 (0)}{5 π}$

Step 5.1.3.1.2

Cancel the common factors.

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Step 5.1.3.1.2.1

Factor $5$ out of $5 π$ .

$\cos (\frac{π x}{12} + \frac{π}{3}) = \frac{5 (0)}{5 (π)}$

Step 5.1.3.1.2.2

Cancel the common factor.

$\cos (\frac{π x}{12} + \frac{π}{3}) = \frac{5 \cdot 0}{5 π}$

Step 5.1.3.1.2.3

Rewrite the expression.

$\cos (\frac{π x}{12} + \frac{π}{3}) = \frac{0}{π}$

Step 5.1.3.2

Divide $0$ by $π$ .

$\cos (\frac{π x}{12} + \frac{π}{3}) = 0$

Step 5.2

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

$\frac{π x}{12} + \frac{π}{3} = \arccos (0)$

Step 5.3

Simplify the right side.

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Step 5.3.1

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

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

Step 5.4

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

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Step 5.4.1

Subtract $\frac{π}{3}$ from both sides of the equation.

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

Step 5.4.2

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

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

Step 5.4.3

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

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

Step 5.4.4

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

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Step 5.4.4.1

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

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

Step 5.4.4.2

Multiply $2$ by $3$ .

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

Step 5.4.4.3

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

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

Step 5.4.4.4

Multiply $3$ by $2$ .

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

Step 5.4.5

Combine the numerators over the common denominator.

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

Step 5.4.6

Simplify the numerator.

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Step 5.4.6.1

Move $3$ to the left of $π$ .

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

Step 5.4.6.2

Multiply $2$ by $- 1$ .

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

Step 5.4.6.3

Subtract $2 π$ from $3 π$ .

$\frac{π x}{12} = \frac{π}{6}$

Step 5.5

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

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

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

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Step 5.6.1.1.1

Cancel the common factor of $12$ .

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Step 5.6.1.1.1.1

Cancel the common factor.

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

Step 5.6.1.1.1.2

Rewrite the expression.

$\frac{1}{π} (π x) = \frac{12}{π} \cdot \frac{π}{6}$

Step 5.6.1.1.2

Cancel the common factor of $π$ .

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Step 5.6.1.1.2.1

Factor $π$ out of $π x$ .

$\frac{1}{π} (π (x)) = \frac{12}{π} \cdot \frac{π}{6}$

Step 5.6.1.1.2.2

Cancel the common factor.

$\frac{1}{π} (π x) = \frac{12}{π} \cdot \frac{π}{6}$

Step 5.6.1.1.2.3

Rewrite the expression.

$x = \frac{12}{π} \cdot \frac{π}{6}$

Step 5.6.2

Simplify the right side.

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Step 5.6.2.1

Simplify $\frac{12}{π} \cdot \frac{π}{6}$ .

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Step 5.6.2.1.1

Cancel the common factor of $6$ .

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Step 5.6.2.1.1.1

Factor $6$ out of $12$ .

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

Step 5.6.2.1.1.2

Cancel the common factor.

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

Step 5.6.2.1.1.3

Rewrite the expression.

$x = \frac{2}{π} π$

Step 5.6.2.1.2

Cancel the common factor of $π$ .

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Step 5.6.2.1.2.1

Cancel the common factor.

$x = \frac{2}{π} π$

Step 5.6.2.1.2.2

Rewrite the expression.

$x = 2$

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.

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

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 $\frac{2}{2}$ .

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

Step 5.8.1.2

Combine fractions.

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Step 5.8.1.2.1

Combine $2 π$ and $\frac{2}{2}$ .

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

Step 5.8.1.2.2

Combine the numerators over the common denominator.

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

Step 5.8.1.3

Simplify the numerator.

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Step 5.8.1.3.1

Multiply $2$ by $2$ .

$\frac{π x}{12} + \frac{π}{3} = \frac{4 π - π}{2}$

Step 5.8.1.3.2

Subtract $π$ from $4 π$ .

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

Step 5.8.2

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

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Step 5.8.2.1

Subtract $\frac{π}{3}$ from both sides of the equation.

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

Step 5.8.2.2

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

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

Step 5.8.2.3

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

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

Step 5.8.2.4

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

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Step 5.8.2.4.1

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

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

Step 5.8.2.4.2

Multiply $2$ by $3$ .

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

Step 5.8.2.4.3

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

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

Step 5.8.2.4.4

Multiply $3$ by $2$ .

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

Step 5.8.2.5

Combine the numerators over the common denominator.

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

Step 5.8.2.6

Simplify the numerator.

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Step 5.8.2.6.1

Multiply $3$ by $3$ .

$\frac{π x}{12} = \frac{9 π - π \cdot 2}{6}$

Step 5.8.2.6.2

Multiply $2$ by $- 1$ .

$\frac{π x}{12} = \frac{9 π - 2 π}{6}$

Step 5.8.2.6.3

Subtract $2 π$ from $9 π$ .

$\frac{π x}{12} = \frac{7 π}{6}$

Step 5.8.3

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

$\frac{12}{π} \cdot \frac{π x}{12} = \frac{12}{π} \cdot \frac{7 π}{6}$

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

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Step 5.8.4.1.1.1

Cancel the common factor of $12$ .

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Step 5.8.4.1.1.1.1

Cancel the common factor.

$\frac{12}{π} \cdot \frac{π x}{12} = \frac{12}{π} \cdot \frac{7 π}{6}$

Step 5.8.4.1.1.1.2

Rewrite the expression.

$\frac{1}{π} (π x) = \frac{12}{π} \cdot \frac{7 π}{6}$

Step 5.8.4.1.1.2

Cancel the common factor of $π$ .

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Step 5.8.4.1.1.2.1

Factor $π$ out of $π x$ .

$\frac{1}{π} (π (x)) = \frac{12}{π} \cdot \frac{7 π}{6}$

Step 5.8.4.1.1.2.2

Cancel the common factor.

$\frac{1}{π} (π x) = \frac{12}{π} \cdot \frac{7 π}{6}$

Step 5.8.4.1.1.2.3

Rewrite the expression.

$x = \frac{12}{π} \cdot \frac{7 π}{6}$

Step 5.8.4.2

Simplify the right side.

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Step 5.8.4.2.1

Simplify $\frac{12}{π} \cdot \frac{7 π}{6}$ .

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Step 5.8.4.2.1.1

Cancel the common factor of $6$ .

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Step 5.8.4.2.1.1.1

Factor $6$ out of $12$ .

$x = \frac{6 (2)}{π} \cdot \frac{7 π}{6}$

Step 5.8.4.2.1.1.2

Cancel the common factor.

$x = \frac{6 \cdot 2}{π} \cdot \frac{7 π}{6}$

Step 5.8.4.2.1.1.3

Rewrite the expression.

$x = \frac{2}{π} (7 π)$

Step 5.8.4.2.1.2

Cancel the common factor of $π$ .

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Step 5.8.4.2.1.2.1

Factor $π$ out of $7 π$ .

$x = \frac{2}{π} (π \cdot 7)$

Step 5.8.4.2.1.2.2

Cancel the common factor.

$x = \frac{2}{π} (π \cdot 7)$

Step 5.8.4.2.1.2.3

Rewrite the expression.

$x = 2 \cdot 7$

Step 5.8.4.2.1.3

Multiply $2$ by $7$ .

$x = 14$

Step 5.9

The solution to the equation $\frac{π x}{12} + \frac{π}{3} = \frac{π}{2}$ .

$x = 2, 14$

Step 6

Evaluate the second derivative at $x = 2$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{5 π^{2} \sin (\frac{π (2)}{12} + \frac{π}{3})}{144}$

Step 7

Evaluate the second derivative.

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Step 7.1

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

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Step 7.1.1

Factor $2$ out of $π (2)$ .

$\frac{5 π^{2} \sin (\frac{2 π}{12} + \frac{π}{3})}{144}$

Step 7.1.2

Cancel the common factors.

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Step 7.1.2.1

Factor $2$ out of $12$ .

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

Step 7.1.2.2

Cancel the common factor.

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

Step 7.1.2.3

Rewrite the expression.

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

Step 7.2

Simplify the numerator.

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Step 7.2.1

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

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

Step 7.2.2

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

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Step 7.2.2.1

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

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

Step 7.2.2.2

Multiply $3$ by $2$ .

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

Step 7.2.3

Combine the numerators over the common denominator.

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

Step 7.2.4

Simplify the numerator.

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Step 7.2.4.1

Move $2$ to the left of $π$ .

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

Step 7.2.4.2

Add $π$ and $2 π$ .

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

Step 7.2.5

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

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Step 7.2.5.1

Factor $3$ out of $3 π$ .

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

Step 7.2.5.2

Cancel the common factors.

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Step 7.2.5.2.1

Factor $3$ out of $6$ .

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

Step 7.2.5.2.2

Cancel the common factor.

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

Step 7.2.5.2.3

Rewrite the expression.

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

Step 7.2.6

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

$\frac{5 π^{2} \cdot 1}{144}$

Step 7.2.7

Multiply $5$ by $1$ .

$\frac{5 π^{2}}{144}$

Step 8

$x = 2$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = 2$ is a local minimum

Step 9

Find the y-value when $x = 2$ .

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Step 9.1

Replace the variable $x$ with $2$ in the expression.

$f (2) = - 5 \sin (\frac{π}{12} \cdot (2) + \frac{π}{3}) + 2$

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

Cancel the common factor of $2$ .

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Step 9.2.1.1.1

Factor $2$ out of $12$ .

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

Step 9.2.1.1.2

Cancel the common factor.

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

Step 9.2.1.1.3

Rewrite the expression.

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

Step 9.2.1.2

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

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

Step 9.2.1.3

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

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Step 9.2.1.3.1

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

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

Step 9.2.1.3.2

Multiply $3$ by $2$ .

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

Step 9.2.1.4

Combine the numerators over the common denominator.

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

Step 9.2.1.5

Simplify the numerator.

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Step 9.2.1.5.1

Move $2$ to the left of $π$ .

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

Step 9.2.1.5.2

Add $π$ and $2 π$ .

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

Step 9.2.1.6

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

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Step 9.2.1.6.1

Factor $3$ out of $3 π$ .

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

Step 9.2.1.6.2

Cancel the common factors.

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Step 9.2.1.6.2.1

Factor $3$ out of $6$ .

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

Step 9.2.1.6.2.2

Cancel the common factor.

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

Step 9.2.1.6.2.3

Rewrite the expression.

$f (2) = - 5 \sin (\frac{π}{2}) + 2$

Step 9.2.1.7

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

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

Step 9.2.1.8

Multiply $- 5$ by $1$ .

$f (2) = - 5 + 2$

Step 9.2.2

Add $- 5$ and $2$ .

$f (2) = - 3$

Step 9.2.3

The final answer is $- 3$ .

$y = - 3$

Step 10

Evaluate the second derivative at $x = 14$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{5 π^{2} \sin (\frac{π (14)}{12} + \frac{π}{3})}{144}$

Step 11

Evaluate the second derivative.

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Step 11.1

Cancel the common factor of $14$ and $12$ .

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Step 11.1.1

Factor $2$ out of $π (14)$ .

$\frac{5 π^{2} \sin (\frac{2 (π \cdot (7))}{12} + \frac{π}{3})}{144}$

Step 11.1.2

Cancel the common factors.

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Step 11.1.2.1

Factor $2$ out of $12$ .

$\frac{5 π^{2} \sin (\frac{2 (π \cdot (7))}{2 (6)} + \frac{π}{3})}{144}$

Step 11.1.2.2

Cancel the common factor.

$\frac{5 π^{2} \sin (\frac{2 (π \cdot (7))}{2 \cdot 6} + \frac{π}{3})}{144}$

Step 11.1.2.3

Rewrite the expression.

$\frac{5 π^{2} \sin (\frac{π \cdot (7)}{6} + \frac{π}{3})}{144}$

Step 11.2

Simplify the numerator.

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Step 11.2.1

Move $7$ to the left of $π$ .

$\frac{5 π^{2} \sin (\frac{7 π}{6} + \frac{π}{3})}{144}$

Step 11.2.2

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

$\frac{5 π^{2} \sin (\frac{7 π}{6} + \frac{π}{3} \cdot \frac{2}{2})}{144}$

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 $\frac{π}{3}$ by $\frac{2}{2}$ .

$\frac{5 π^{2} \sin (\frac{7 π}{6} + \frac{π \cdot 2}{3 \cdot 2})}{144}$

Step 11.2.3.2

Multiply $3$ by $2$ .

$\frac{5 π^{2} \sin (\frac{7 π}{6} + \frac{π \cdot 2}{6})}{144}$

Step 11.2.4

Combine the numerators over the common denominator.

$\frac{5 π^{2} \sin (\frac{7 π + π \cdot 2}{6})}{144}$

Step 11.2.5

Simplify the numerator.

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Step 11.2.5.1

Move $2$ to the left of $π$ .

$\frac{5 π^{2} \sin (\frac{7 π + 2 \cdot π}{6})}{144}$

Step 11.2.5.2

Add $7 π$ and $2 π$ .

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

Step 11.2.6

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

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Step 11.2.6.1

Factor $3$ out of $9 π$ .

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

Step 11.2.6.2

Cancel the common factors.

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Step 11.2.6.2.1

Factor $3$ out of $6$ .

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

Step 11.2.6.2.2

Cancel the common factor.

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

Step 11.2.6.2.3

Rewrite the expression.

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

Step 11.2.7

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

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

Step 11.2.8

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

$\frac{5 π^{2} (- 1 \cdot 1)}{144}$

Step 11.2.9

Multiply $- 1$ by $1$ .

$\frac{5 π^{2} \cdot - 1}{144}$

Step 11.2.10

Multiply $- 1$ by $5$ .

$\frac{- 5 π^{2}}{144}$

Step 11.3

Move the negative in front of the fraction.

$- \frac{5 π^{2}}{144}$

Step 12

$x = 14$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = 14$ is a local maximum

Step 13

Find the y-value when $x = 14$ .

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Step 13.1

Replace the variable $x$ with $14$ in the expression.

$f (14) = - 5 \sin (\frac{π}{12} \cdot (14) + \frac{π}{3}) + 2$

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 $2$ .

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Step 13.2.1.1.1.1

Factor $2$ out of $12$ .

$f (14) = - 5 \sin (\frac{π}{2 (6)} \cdot 14 + \frac{π}{3}) + 2$

Step 13.2.1.1.1.2

Factor $2$ out of $14$ .

$f (14) = - 5 \sin (\frac{π}{2 \cdot 6} \cdot (2 \cdot 7) + \frac{π}{3}) + 2$

Step 13.2.1.1.1.3

Cancel the common factor.

$f (14) = - 5 \sin (\frac{π}{2 \cdot 6} \cdot (2 \cdot 7) + \frac{π}{3}) + 2$

Step 13.2.1.1.1.4

Rewrite the expression.

$f (14) = - 5 \sin (\frac{π}{6} \cdot 7 + \frac{π}{3}) + 2$

Step 13.2.1.1.2

Combine $\frac{π}{6}$ and $7$ .

$f (14) = - 5 \sin (\frac{π \cdot 7}{6} + \frac{π}{3}) + 2$

Step 13.2.1.1.3

Move $7$ to the left of $π$ .

$f (14) = - 5 \sin (\frac{7 π}{6} + \frac{π}{3}) + 2$

Step 13.2.1.2

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

$f (14) = - 5 \sin (\frac{7 π}{6} + \frac{π}{3} \cdot \frac{2}{2}) + 2$

Step 13.2.1.3

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

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Step 13.2.1.3.1

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

$f (14) = - 5 \sin (\frac{7 π}{6} + \frac{π \cdot 2}{3 \cdot 2}) + 2$

Step 13.2.1.3.2

Multiply $3$ by $2$ .

$f (14) = - 5 \sin (\frac{7 π}{6} + \frac{π \cdot 2}{6}) + 2$

Step 13.2.1.4

Combine the numerators over the common denominator.

$f (14) = - 5 \sin (\frac{7 π + π \cdot 2}{6}) + 2$

Step 13.2.1.5

Simplify the numerator.

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Step 13.2.1.5.1

Move $2$ to the left of $π$ .

$f (14) = - 5 \sin (\frac{7 π + 2 \cdot π}{6}) + 2$

Step 13.2.1.5.2

Add $7 π$ and $2 π$ .

$f (14) = - 5 \sin (\frac{9 π}{6}) + 2$

Step 13.2.1.6

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

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Step 13.2.1.6.1

Factor $3$ out of $9 π$ .

$f (14) = - 5 \sin (\frac{3 (3 π)}{6}) + 2$

Step 13.2.1.6.2

Cancel the common factors.

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Step 13.2.1.6.2.1

Factor $3$ out of $6$ .

$f (14) = - 5 \sin (\frac{3 (3 π)}{3 (2)}) + 2$

Step 13.2.1.6.2.2

Cancel the common factor.

$f (14) = - 5 \sin (\frac{3 (3 π)}{3 \cdot 2}) + 2$

Step 13.2.1.6.2.3

Rewrite the expression.

$f (14) = - 5 \sin (\frac{3 π}{2}) + 2$

Step 13.2.1.7

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.

$f (14) = - 5 (- \sin (\frac{π}{2})) + 2$

Step 13.2.1.8

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

$f (14) = - 5 (- 1 \cdot 1) + 2$

Step 13.2.1.9

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

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Step 13.2.1.9.1

Multiply $- 1$ by $1$ .

$f (14) = - 5 \cdot - 1 + 2$

Step 13.2.1.9.2

Multiply $- 5$ by $- 1$ .

$f (14) = 5 + 2$

Step 13.2.2

Add $5$ and $2$ .

$f (14) = 7$

Step 13.2.3

The final answer is $7$ .

$y = 7$

Step 14

These are the local extrema for $f (x) = - 5 \sin (\frac{π}{12} x + \frac{π}{3}) + 2$ .

$(2, - 3)$ is a local minima

$(14, 7)$ is a local maxima

Step 15