Calculus Examples

$f (x) = 3 \sin^{2} (x) + 3 \sin (x)$

Step 1

Find the first derivative of the function.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

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

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

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 $f (x) = x^{2}$ and $g (x) = \sin (x)$ .

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Step 1.2.2.1

To apply the Chain Rule, set $u$ as $\sin (x)$ .

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

Step 1.2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 2$ .

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

Step 1.2.2.3

Replace all occurrences of $u$ with $\sin (x)$ .

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

Step 1.2.3

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

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

Step 1.2.4

Multiply $2$ by $3$ .

$6 \sin (x) \cos (x) + \frac{d}{d x} [3 \sin (x)]$

Step 1.3

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

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Step 1.3.1

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

$6 \sin (x) \cos (x) + 3 \frac{d}{d x} [\sin (x)]$

Step 1.3.2

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

$6 \sin (x) \cos (x) + 3 \cos (x)$

Step 1.4

Reorder terms.

$6 \cos (x) \sin (x) + 3 \cos (x)$

Step 2

Find the second derivative of the function.

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Step 2.1

By the Sum Rule, the derivative of $6 \cos (x) \sin (x) + 3 \cos (x)$ with respect to $x$ is $\frac{d}{d x} [6 \cos (x) \sin (x)] + \frac{d}{d x} [3 \cos (x)]$ .

$f'' (x) = \frac{d}{d x} (6 \cos (x) \sin (x)) + \frac{d}{d x} (3 \cos (x))$

Step 2.2

Evaluate $\frac{d}{d x} [6 \cos (x) \sin (x)]$ .

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Step 2.2.1

Since $6$ is constant with respect to $x$ , the derivative of $6 \cos (x) \sin (x)$ with respect to $x$ is $6 \frac{d}{d x} [\cos (x) \sin (x)]$ .

$f'' (x) = 6 \frac{d}{d x} (\cos (x) \sin (x)) + \frac{d}{d x} (3 \cos (x))$

Step 2.2.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = \cos (x)$ and $g (x) = \sin (x)$ .

$f'' (x) = 6 (\cos (x) \frac{d}{d x} (\sin (x)) + \sin (x) \frac{d}{d x} (\cos (x))) + \frac{d}{d x} (3 \cos (x))$

Step 2.2.3

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

$f'' (x) = 6 (\cos (x) \cos (x) + \sin (x) \frac{d}{d x} (\cos (x))) + \frac{d}{d x} (3 \cos (x))$

Step 2.2.4

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

$f'' (x) = 6 (\cos (x) \cos (x) + \sin (x) (- \sin (x))) + \frac{d}{d x} (3 \cos (x))$

Step 2.2.5

Raise $\cos (x)$ to the power of $1$ .

$f'' (x) = 6 (\cos (x) \cos (x) + \sin (x) (- \sin (x))) + \frac{d}{d x} (3 \cos (x))$

Step 2.2.6

Raise $\cos (x)$ to the power of $1$ .

$f'' (x) = 6 (\cos (x) \cos (x) + \sin (x) (- \sin (x))) + \frac{d}{d x} (3 \cos (x))$

Step 2.2.7

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

$f'' (x) = 6 ({\cos (x)}^{1 + 1} + \sin (x) (- \sin (x))) + \frac{d}{d x} (3 \cos (x))$

Step 2.2.8

Add $1$ and $1$ .

$f'' (x) = 6 (\cos^{2} (x) + \sin (x) (- \sin (x))) + \frac{d}{d x} (3 \cos (x))$

Step 2.2.9

Raise $\sin (x)$ to the power of $1$ .

$f'' (x) = 6 (\cos^{2} (x) - (\sin (x) \sin (x))) + \frac{d}{d x} (3 \cos (x))$

Step 2.2.10

Raise $\sin (x)$ to the power of $1$ .

$f'' (x) = 6 (\cos^{2} (x) - (\sin (x) \sin (x))) + \frac{d}{d x} (3 \cos (x))$

Step 2.2.11

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

$f'' (x) = 6 (\cos^{2} (x) - {\sin (x)}^{1 + 1}) + \frac{d}{d x} (3 \cos (x))$

Step 2.2.12

Add $1$ and $1$ .

$f'' (x) = 6 (\cos^{2} (x) - \sin^{2} (x)) + \frac{d}{d x} (3 \cos (x))$

Step 2.3

Evaluate $\frac{d}{d x} [3 \cos (x)]$ .

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Step 2.3.1

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

$f'' (x) = 6 (\cos^{2} (x) - \sin^{2} (x)) + 3 \frac{d}{d x} (\cos (x))$

Step 2.3.2

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

$f'' (x) = 6 (\cos^{2} (x) - \sin^{2} (x)) + 3 (- \sin (x))$

Step 2.3.3

Multiply $- 1$ by $3$ .

$f'' (x) = 6 (\cos^{2} (x) - \sin^{2} (x)) - 3 \sin (x)$

Step 2.4

Simplify.

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Step 2.4.1

Apply the distributive property.

$f'' (x) = 6 \cos^{2} (x) + 6 (- \sin^{2} (x)) - 3 \sin (x)$

Step 2.4.2

Multiply $- 1$ by $6$ .

$f'' (x) = 6 \cos^{2} (x) - 6 \sin^{2} (x) - 3 \sin (x)$

Step 3

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

$6 \cos (x) \sin (x) + 3 \cos (x) = 0$

Step 4

Factor $3 \cos (x)$ out of $6 \cos (x) \sin (x) + 3 \cos (x)$ .

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Step 4.1

Factor $3 \cos (x)$ out of $6 \cos (x) \sin (x)$ .

$3 \cos (x) (2 \sin (x)) + 3 \cos (x) = 0$

Step 4.2

Factor $3 \cos (x)$ out of $3 \cos (x)$ .

$3 \cos (x) (2 \sin (x)) + 3 \cos (x) \cdot 1 = 0$

Step 4.3

Factor $3 \cos (x)$ out of $3 \cos (x) (2 \sin (x)) + 3 \cos (x) \cdot 1$ .

$3 \cos (x) (2 \sin (x) + 1) = 0$

Step 5

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$\cos (x) = 0$

$2 \sin (x) + 1 = 0$

Step 6

Set $\cos (x)$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $\cos (x)$ equal to $0$ .

$\cos (x) = 0$

Step 6.2

Solve $\cos (x) = 0$ for $x$ .

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Step 6.2.1

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

$x = \arccos (0)$

Step 6.2.2

Simplify the right side.

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Step 6.2.2.1

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

$x = \frac{π}{2}$

Step 6.2.3

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.

$x = 2 π - \frac{π}{2}$

Step 6.2.4

Simplify $2 π - \frac{π}{2}$ .

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Step 6.2.4.1

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

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

Step 6.2.4.2

Combine fractions.

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Step 6.2.4.2.1

Combine $2 π$ and $\frac{2}{2}$ .

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

Step 6.2.4.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 2 - π}{2}$

Step 6.2.4.3

Simplify the numerator.

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Step 6.2.4.3.1

Multiply $2$ by $2$ .

$x = \frac{4 π - π}{2}$

Step 6.2.4.3.2

Subtract $π$ from $4 π$ .

$x = \frac{3 π}{2}$

Step 6.2.5

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

$x = \frac{π}{2}, \frac{3 π}{2}$

Step 7

Set $2 \sin (x) + 1$ equal to $0$ and solve for $x$ .

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Step 7.1

Set $2 \sin (x) + 1$ equal to $0$ .

$2 \sin (x) + 1 = 0$

Step 7.2

Solve $2 \sin (x) + 1 = 0$ for $x$ .

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Step 7.2.1

Subtract $1$ from both sides of the equation.

$2 \sin (x) = - 1$

Step 7.2.2

Divide each term in $2 \sin (x) = - 1$ by $2$ and simplify.

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Step 7.2.2.1

Divide each term in $2 \sin (x) = - 1$ by $2$ .

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

Step 7.2.2.2

Simplify the left side.

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Step 7.2.2.2.1

Cancel the common factor of $2$ .

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Step 7.2.2.2.1.1

Cancel the common factor.

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

Step 7.2.2.2.1.2

Divide $\sin (x)$ by $1$ .

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

Step 7.2.2.3

Simplify the right side.

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Step 7.2.2.3.1

Move the negative in front of the fraction.

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

Step 7.2.3

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

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

Step 7.2.4

Simplify the right side.

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Step 7.2.4.1

The exact value of $\arcsin (- \frac{1}{2})$ is $- \frac{π}{6}$ .

$x = - \frac{π}{6}$

Step 7.2.5

The sine function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from $2 π$ , to find a reference angle. Next, add this reference angle to $π$ to find the solution in the third quadrant.

$x = 2 π + \frac{π}{6} + π$

Step 7.2.6

Simplify the expression to find the second solution.

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Step 7.2.6.1

Subtract $2 π$ from $2 π + \frac{π}{6} + π$ .

$x = 2 π + \frac{π}{6} + π - 2 π$

Step 7.2.6.2

The resulting angle of $\frac{7 π}{6}$ is positive, less than $2 π$ , and coterminal with $2 π + \frac{π}{6} + π$ .

$x = \frac{7 π}{6}$

Step 7.2.7

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

$x = - \frac{π}{6}, \frac{7 π}{6}$

Step 8

The final solution is all the values that make $3 \cos (x) (2 \sin (x) + 1) = 0$ true.

$x = \frac{π}{2}, \frac{3 π}{2}, - \frac{π}{6}, \frac{7 π}{6}$

Step 9

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

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

Step 10

Evaluate the second derivative.

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Step 10.1

Simplify each term.

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Step 10.1.1

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

$6 \cdot 0^{2} - 6 \sin^{2} (\frac{π}{2}) - 3 \sin (\frac{π}{2})$

Step 10.1.2

Raising $0$ to any positive power yields $0$ .

$6 \cdot 0 - 6 \sin^{2} (\frac{π}{2}) - 3 \sin (\frac{π}{2})$

Step 10.1.3

Multiply $6$ by $0$ .

$0 - 6 \sin^{2} (\frac{π}{2}) - 3 \sin (\frac{π}{2})$

Step 10.1.4

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

$0 - 6 \cdot 1^{2} - 3 \sin (\frac{π}{2})$

Step 10.1.5

One to any power is one.

$0 - 6 \cdot 1 - 3 \sin (\frac{π}{2})$

Step 10.1.6

Multiply $- 6$ by $1$ .

$0 - 6 - 3 \sin (\frac{π}{2})$

Step 10.1.7

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

$0 - 6 - 3 \cdot 1$

Step 10.1.8

Multiply $- 3$ by $1$ .

$0 - 6 - 3$

Step 10.2

Simplify by subtracting numbers.

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Step 10.2.1

Subtract $6$ from $0$ .

$- 6 - 3$

Step 10.2.2

Subtract $3$ from $- 6$ .

$- 9$

Step 11

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

$x = \frac{π}{2}$ is a local maximum

Step 12

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

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Step 12.1

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

$f (\frac{π}{2}) = 3 \sin^{2} (\frac{π}{2}) + 3 \sin (\frac{π}{2})$

Step 12.2

Simplify the result.

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Step 12.2.1

Simplify each term.

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Step 12.2.1.1

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

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

Step 12.2.1.2

One to any power is one.

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

Step 12.2.1.3

Multiply $3$ by $1$ .

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

Step 12.2.1.4

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

$f (\frac{π}{2}) = 3 + 3 \cdot 1$

Step 12.2.1.5

Multiply $3$ by $1$ .

$f (\frac{π}{2}) = 3 + 3$

Step 12.2.2

Add $3$ and $3$ .

$f (\frac{π}{2}) = 6$

Step 12.2.3

The final answer is $6$ .

$y = 6$

Step 13

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

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

Step 14

Evaluate the second derivative.

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Step 14.1

Simplify each term.

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Step 14.1.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

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

Step 14.1.2

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

$6 \cdot 0^{2} - 6 \sin^{2} (\frac{3 π}{2}) - 3 \sin (\frac{3 π}{2})$

Step 14.1.3

Raising $0$ to any positive power yields $0$ .

$6 \cdot 0 - 6 \sin^{2} (\frac{3 π}{2}) - 3 \sin (\frac{3 π}{2})$

Step 14.1.4

Multiply $6$ by $0$ .

$0 - 6 \sin^{2} (\frac{3 π}{2}) - 3 \sin (\frac{3 π}{2})$

Step 14.1.5

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.

$0 - 6 {(- \sin (\frac{π}{2}))}^{2} - 3 \sin (\frac{3 π}{2})$

Step 14.1.6

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

$0 - 6 {(- 1 \cdot 1)}^{2} - 3 \sin (\frac{3 π}{2})$

Step 14.1.7

Multiply $- 1$ by $1$ .

$0 - 6 {(- 1)}^{2} - 3 \sin (\frac{3 π}{2})$

Step 14.1.8

Raise $- 1$ to the power of $2$ .

$0 - 6 \cdot 1 - 3 \sin (\frac{3 π}{2})$

Step 14.1.9

Multiply $- 6$ by $1$ .

$0 - 6 - 3 \sin (\frac{3 π}{2})$

Step 14.1.10

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.

$0 - 6 - 3 (- \sin (\frac{π}{2}))$

Step 14.1.11

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

$0 - 6 - 3 (- 1 \cdot 1)$

Step 14.1.12

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

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Step 14.1.12.1

Multiply $- 1$ by $1$ .

$0 - 6 - 3 \cdot - 1$

Step 14.1.12.2

Multiply $- 3$ by $- 1$ .

$0 - 6 + 3$

Step 14.2

Simplify by adding and subtracting.

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Step 14.2.1

Subtract $6$ from $0$ .

$- 6 + 3$

Step 14.2.2

Add $- 6$ and $3$ .

$- 3$

Step 15

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

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

Step 16

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

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Step 16.1

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

$f (\frac{3 π}{2}) = 3 \sin^{2} (\frac{3 π}{2}) + 3 \sin (\frac{3 π}{2})$

Step 16.2

Simplify the result.

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Step 16.2.1

Simplify each term.

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Step 16.2.1.1

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.

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

Step 16.2.1.2

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

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

Step 16.2.1.3

Multiply $- 1$ by $1$ .

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

Step 16.2.1.4

Raise $- 1$ to the power of $2$ .

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

Step 16.2.1.5

Multiply $3$ by $1$ .

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

Step 16.2.1.6

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

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

Step 16.2.1.7

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

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

Step 16.2.1.8

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

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Step 16.2.1.8.1

Multiply $- 1$ by $1$ .

$f (\frac{3 π}{2}) = 3 + 3 \cdot - 1$

Step 16.2.1.8.2

Multiply $3$ by $- 1$ .

$f (\frac{3 π}{2}) = 3 - 3$

Step 16.2.2

Subtract $3$ from $3$ .

$f (\frac{3 π}{2}) = 0$

Step 16.2.3

The final answer is $0$ .

$y = 0$

Step 17

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

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

Step 18

Evaluate the second derivative.

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Step 18.1

Simplify each term.

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Step 18.1.1

Add full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$6 \cos^{2} (\frac{11 π}{6}) - 6 \sin^{2} (- \frac{π}{6}) - 3 \sin (- \frac{π}{6})$

Step 18.1.2

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

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

Step 18.1.3

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

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

Step 18.1.4

Apply the product rule to $\frac{\sqrt{3}}{2}$ .

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

Step 18.1.5

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 18.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

Step 18.1.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 18.1.5.3

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

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

Step 18.1.5.4

Cancel the common factor of $2$ .

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Step 18.1.5.4.1

Cancel the common factor.

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

Step 18.1.5.4.2

Rewrite the expression.

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

Step 18.1.5.5

Evaluate the exponent.

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

Step 18.1.6

Raise $2$ to the power of $2$ .

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

Step 18.1.7

Cancel the common factor of $2$ .

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Step 18.1.7.1

Factor $2$ out of $6$ .

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

Step 18.1.7.2

Factor $2$ out of $4$ .

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

Step 18.1.7.3

Cancel the common factor.

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

Step 18.1.7.4

Rewrite the expression.

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

Step 18.1.8

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

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

Step 18.1.9

Multiply $3$ by $3$ .

$\frac{9}{2} - 6 \sin^{2} (- \frac{π}{6}) - 3 \sin (- \frac{π}{6})$

Step 18.1.10

Add full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\frac{9}{2} - 6 \sin^{2} (\frac{11 π}{6}) - 3 \sin (- \frac{π}{6})$

Step 18.1.11

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.

$\frac{9}{2} - 6 {(- \sin (\frac{π}{6}))}^{2} - 3 \sin (- \frac{π}{6})$

Step 18.1.12

The exact value of $\sin (\frac{π}{6})$ is $\frac{1}{2}$ .

$\frac{9}{2} - 6 {(- \frac{1}{2})}^{2} - 3 \sin (- \frac{π}{6})$

Step 18.1.13

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 18.1.13.1

Apply the product rule to $- \frac{1}{2}$ .

$\frac{9}{2} - 6 ({(- 1)}^{2} {(\frac{1}{2})}^{2}) - 3 \sin (- \frac{π}{6})$

Step 18.1.13.2

Apply the product rule to $\frac{1}{2}$ .

$\frac{9}{2} - 6 ({(- 1)}^{2} \frac{1^{2}}{2^{2}}) - 3 \sin (- \frac{π}{6})$

Step 18.1.14

Raise $- 1$ to the power of $2$ .

$\frac{9}{2} - 6 (1 \frac{1^{2}}{2^{2}}) - 3 \sin (- \frac{π}{6})$

Step 18.1.15

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

$\frac{9}{2} - 6 \frac{1^{2}}{2^{2}} - 3 \sin (- \frac{π}{6})$

Step 18.1.16

One to any power is one.

$\frac{9}{2} - 6 \frac{1}{2^{2}} - 3 \sin (- \frac{π}{6})$

Step 18.1.17

Raise $2$ to the power of $2$ .

$\frac{9}{2} - 6 (\frac{1}{4}) - 3 \sin (- \frac{π}{6})$

Step 18.1.18

Cancel the common factor of $2$ .

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Step 18.1.18.1

Factor $2$ out of $- 6$ .

$\frac{9}{2} + 2 (- 3) \frac{1}{4} - 3 \sin (- \frac{π}{6})$

Step 18.1.18.2

Factor $2$ out of $4$ .

$\frac{9}{2} + 2 \cdot - 3 \frac{1}{2 \cdot 2} - 3 \sin (- \frac{π}{6})$

Step 18.1.18.3

Cancel the common factor.

$\frac{9}{2} + 2 \cdot - 3 \frac{1}{2 \cdot 2} - 3 \sin (- \frac{π}{6})$

Step 18.1.18.4

Rewrite the expression.

$\frac{9}{2} - 3 (\frac{1}{2}) - 3 \sin (- \frac{π}{6})$

Step 18.1.19

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

$\frac{9}{2} + \frac{- 3}{2} - 3 \sin (- \frac{π}{6})$

Step 18.1.20

Move the negative in front of the fraction.

$\frac{9}{2} - \frac{3}{2} - 3 \sin (- \frac{π}{6})$

Step 18.1.21

Add full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\frac{9}{2} - \frac{3}{2} - 3 \sin (\frac{11 π}{6})$

Step 18.1.22

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.

$\frac{9}{2} - \frac{3}{2} - 3 (- \sin (\frac{π}{6}))$

Step 18.1.23

The exact value of $\sin (\frac{π}{6})$ is $\frac{1}{2}$ .

$\frac{9}{2} - \frac{3}{2} - 3 (- \frac{1}{2})$

Step 18.1.24

Multiply $- 3 (- \frac{1}{2})$ .

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Step 18.1.24.1

Multiply $- 1$ by $- 3$ .

$\frac{9}{2} - \frac{3}{2} + 3 (\frac{1}{2})$

Step 18.1.24.2

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

$\frac{9}{2} - \frac{3}{2} + \frac{3}{2}$

Step 18.2

Combine fractions.

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Step 18.2.1

Combine the numerators over the common denominator.

$\frac{9 - 3 + 3}{2}$

Step 18.2.2

Simplify by adding and subtracting.

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Step 18.2.2.1

Subtract $3$ from $9$ .

$\frac{6 + 3}{2}$

Step 18.2.2.2

Add $6$ and $3$ .

$\frac{9}{2}$

Step 19

$x = - \frac{π}{6}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = - \frac{π}{6}$ is a local minimum

Step 20

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

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Step 20.1

Replace the variable $x$ with $- \frac{π}{6}$ in the expression.

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

Step 20.2

Simplify the result.

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Step 20.2.1

Simplify each term.

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Step 20.2.1.1

Add full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$f (- \frac{π}{6}) = 3 \sin^{2} (\frac{11 π}{6}) + 3 \sin (- \frac{π}{6})$

Step 20.2.1.2

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.

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

Step 20.2.1.3

The exact value of $\sin (\frac{π}{6})$ is $\frac{1}{2}$ .

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

Step 20.2.1.4

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 20.2.1.4.1

Apply the product rule to $- \frac{1}{2}$ .

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

Step 20.2.1.4.2

Apply the product rule to $\frac{1}{2}$ .

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

Step 20.2.1.5

Raise $- 1$ to the power of $2$ .

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

Step 20.2.1.6

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

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

Step 20.2.1.7

One to any power is one.

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

Step 20.2.1.8

Raise $2$ to the power of $2$ .

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

Step 20.2.1.9

Combine $3$ and $\frac{1}{4}$ .

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

Step 20.2.1.10

Add full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$f (- \frac{π}{6}) = \frac{3}{4} + 3 \sin (\frac{11 π}{6})$

Step 20.2.1.11

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.

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

Step 20.2.1.12

The exact value of $\sin (\frac{π}{6})$ is $\frac{1}{2}$ .

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

Step 20.2.1.13

Multiply $3 (- \frac{1}{2})$ .

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Step 20.2.1.13.1

Multiply $- 1$ by $3$ .

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

Step 20.2.1.13.2

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

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

Step 20.2.1.14

Move the negative in front of the fraction.

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

Step 20.2.2

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

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

Step 20.2.3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 20.2.3.1

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

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

Step 20.2.3.2

Multiply $2$ by $2$ .

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

Step 20.2.4

Combine the numerators over the common denominator.

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

Step 20.2.5

Simplify the numerator.

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Step 20.2.5.1

Multiply $- 3$ by $2$ .

$f (- \frac{π}{6}) = \frac{3 - 6}{4}$

Step 20.2.5.2

Subtract $6$ from $3$ .

$f (- \frac{π}{6}) = \frac{- 3}{4}$

Step 20.2.6

Move the negative in front of the fraction.

$f (- \frac{π}{6}) = - \frac{3}{4}$

Step 20.2.7

The final answer is $- \frac{3}{4}$ .

$y = - \frac{3}{4}$

Step 21

Evaluate the second derivative at $x = \frac{7 π}{6}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$6 \cos^{2} (\frac{7 π}{6}) - 6 \sin^{2} (\frac{7 π}{6}) - 3 \sin (\frac{7 π}{6})$

Step 22

Evaluate the second derivative.

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Step 22.1

Simplify each term.

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Step 22.1.1

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 third quadrant.

$6 {(- \cos (\frac{π}{6}))}^{2} - 6 \sin^{2} (\frac{7 π}{6}) - 3 \sin (\frac{7 π}{6})$

Step 22.1.2

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

$6 {(- \frac{\sqrt{3}}{2})}^{2} - 6 \sin^{2} (\frac{7 π}{6}) - 3 \sin (\frac{7 π}{6})$

Step 22.1.3

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 22.1.3.1

Apply the product rule to $- \frac{\sqrt{3}}{2}$ .

$6 ({(- 1)}^{2} {(\frac{\sqrt{3}}{2})}^{2}) - 6 \sin^{2} (\frac{7 π}{6}) - 3 \sin (\frac{7 π}{6})$

Step 22.1.3.2

Apply the product rule to $\frac{\sqrt{3}}{2}$ .

$6 ({(- 1)}^{2} \frac{{\sqrt{3}}^{2}}{2^{2}}) - 6 \sin^{2} (\frac{7 π}{6}) - 3 \sin (\frac{7 π}{6})$

Step 22.1.4

Raise $- 1$ to the power of $2$ .

$6 (1 \frac{{\sqrt{3}}^{2}}{2^{2}}) - 6 \sin^{2} (\frac{7 π}{6}) - 3 \sin (\frac{7 π}{6})$

Step 22.1.5

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

$6 \frac{{\sqrt{3}}^{2}}{2^{2}} - 6 \sin^{2} (\frac{7 π}{6}) - 3 \sin (\frac{7 π}{6})$

Step 22.1.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 22.1.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$6 \frac{{(3^{\frac{1}{2}})}^{2}}{2^{2}} - 6 \sin^{2} (\frac{7 π}{6}) - 3 \sin (\frac{7 π}{6})$

Step 22.1.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$6 \frac{3^{\frac{1}{2} \cdot 2}}{2^{2}} - 6 \sin^{2} (\frac{7 π}{6}) - 3 \sin (\frac{7 π}{6})$

Step 22.1.6.3

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

$6 \frac{3^{\frac{2}{2}}}{2^{2}} - 6 \sin^{2} (\frac{7 π}{6}) - 3 \sin (\frac{7 π}{6})$

Step 22.1.6.4

Cancel the common factor of $2$ .

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Step 22.1.6.4.1

Cancel the common factor.

$6 \frac{3^{\frac{2}{2}}}{2^{2}} - 6 \sin^{2} (\frac{7 π}{6}) - 3 \sin (\frac{7 π}{6})$

Step 22.1.6.4.2

Rewrite the expression.

$6 \frac{3^{1}}{2^{2}} - 6 \sin^{2} (\frac{7 π}{6}) - 3 \sin (\frac{7 π}{6})$

Step 22.1.6.5

Evaluate the exponent.

$6 \frac{3}{2^{2}} - 6 \sin^{2} (\frac{7 π}{6}) - 3 \sin (\frac{7 π}{6})$

Step 22.1.7

Raise $2$ to the power of $2$ .

$6 (\frac{3}{4}) - 6 \sin^{2} (\frac{7 π}{6}) - 3 \sin (\frac{7 π}{6})$

Step 22.1.8

Cancel the common factor of $2$ .

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Step 22.1.8.1

Factor $2$ out of $6$ .

$2 (3) \frac{3}{4} - 6 \sin^{2} (\frac{7 π}{6}) - 3 \sin (\frac{7 π}{6})$

Step 22.1.8.2

Factor $2$ out of $4$ .

$2 \cdot 3 \frac{3}{2 \cdot 2} - 6 \sin^{2} (\frac{7 π}{6}) - 3 \sin (\frac{7 π}{6})$

Step 22.1.8.3

Cancel the common factor.

$2 \cdot 3 \frac{3}{2 \cdot 2} - 6 \sin^{2} (\frac{7 π}{6}) - 3 \sin (\frac{7 π}{6})$

Step 22.1.8.4

Rewrite the expression.

$3 (\frac{3}{2}) - 6 \sin^{2} (\frac{7 π}{6}) - 3 \sin (\frac{7 π}{6})$

Step 22.1.9

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

$\frac{3 \cdot 3}{2} - 6 \sin^{2} (\frac{7 π}{6}) - 3 \sin (\frac{7 π}{6})$

Step 22.1.10

Multiply $3$ by $3$ .

$\frac{9}{2} - 6 \sin^{2} (\frac{7 π}{6}) - 3 \sin (\frac{7 π}{6})$

Step 22.1.11

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 third quadrant.

$\frac{9}{2} - 6 {(- \sin (\frac{π}{6}))}^{2} - 3 \sin (\frac{7 π}{6})$

Step 22.1.12

The exact value of $\sin (\frac{π}{6})$ is $\frac{1}{2}$ .

$\frac{9}{2} - 6 {(- \frac{1}{2})}^{2} - 3 \sin (\frac{7 π}{6})$

Step 22.1.13

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 22.1.13.1

Apply the product rule to $- \frac{1}{2}$ .

$\frac{9}{2} - 6 ({(- 1)}^{2} {(\frac{1}{2})}^{2}) - 3 \sin (\frac{7 π}{6})$

Step 22.1.13.2

Apply the product rule to $\frac{1}{2}$ .

$\frac{9}{2} - 6 ({(- 1)}^{2} \frac{1^{2}}{2^{2}}) - 3 \sin (\frac{7 π}{6})$

Step 22.1.14

Raise $- 1$ to the power of $2$ .

$\frac{9}{2} - 6 (1 \frac{1^{2}}{2^{2}}) - 3 \sin (\frac{7 π}{6})$

Step 22.1.15

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

$\frac{9}{2} - 6 \frac{1^{2}}{2^{2}} - 3 \sin (\frac{7 π}{6})$

Step 22.1.16

One to any power is one.

$\frac{9}{2} - 6 \frac{1}{2^{2}} - 3 \sin (\frac{7 π}{6})$

Step 22.1.17

Raise $2$ to the power of $2$ .

$\frac{9}{2} - 6 (\frac{1}{4}) - 3 \sin (\frac{7 π}{6})$

Step 22.1.18

Cancel the common factor of $2$ .

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Step 22.1.18.1

Factor $2$ out of $- 6$ .

$\frac{9}{2} + 2 (- 3) \frac{1}{4} - 3 \sin (\frac{7 π}{6})$

Step 22.1.18.2

Factor $2$ out of $4$ .

$\frac{9}{2} + 2 \cdot - 3 \frac{1}{2 \cdot 2} - 3 \sin (\frac{7 π}{6})$

Step 22.1.18.3

Cancel the common factor.

$\frac{9}{2} + 2 \cdot - 3 \frac{1}{2 \cdot 2} - 3 \sin (\frac{7 π}{6})$

Step 22.1.18.4

Rewrite the expression.

$\frac{9}{2} - 3 (\frac{1}{2}) - 3 \sin (\frac{7 π}{6})$

Step 22.1.19

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

$\frac{9}{2} + \frac{- 3}{2} - 3 \sin (\frac{7 π}{6})$

Step 22.1.20

Move the negative in front of the fraction.

$\frac{9}{2} - \frac{3}{2} - 3 \sin (\frac{7 π}{6})$

Step 22.1.21

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 third quadrant.

$\frac{9}{2} - \frac{3}{2} - 3 (- \sin (\frac{π}{6}))$

Step 22.1.22

The exact value of $\sin (\frac{π}{6})$ is $\frac{1}{2}$ .

$\frac{9}{2} - \frac{3}{2} - 3 (- \frac{1}{2})$

Step 22.1.23

Multiply $- 3 (- \frac{1}{2})$ .

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Step 22.1.23.1

Multiply $- 1$ by $- 3$ .

$\frac{9}{2} - \frac{3}{2} + 3 (\frac{1}{2})$

Step 22.1.23.2

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

$\frac{9}{2} - \frac{3}{2} + \frac{3}{2}$

Step 22.2

Combine fractions.

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Step 22.2.1

Combine the numerators over the common denominator.

$\frac{9 - 3 + 3}{2}$

Step 22.2.2

Simplify by adding and subtracting.

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Step 22.2.2.1

Subtract $3$ from $9$ .

$\frac{6 + 3}{2}$

Step 22.2.2.2

Add $6$ and $3$ .

$\frac{9}{2}$

Step 23

$x = \frac{7 π}{6}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = \frac{7 π}{6}$ is a local minimum

Step 24

Find the y-value when $x = \frac{7 π}{6}$ .

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Step 24.1

Replace the variable $x$ with $\frac{7 π}{6}$ in the expression.

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

Step 24.2

Simplify the result.

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Step 24.2.1

Simplify each term.

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Step 24.2.1.1

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 third quadrant.

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

Step 24.2.1.2

The exact value of $\sin (\frac{π}{6})$ is $\frac{1}{2}$ .

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

Step 24.2.1.3

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 24.2.1.3.1

Apply the product rule to $- \frac{1}{2}$ .

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

Step 24.2.1.3.2

Apply the product rule to $\frac{1}{2}$ .

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

Step 24.2.1.4

Raise $- 1$ to the power of $2$ .

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

Step 24.2.1.5

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

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

Step 24.2.1.6

One to any power is one.

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

Step 24.2.1.7

Raise $2$ to the power of $2$ .

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

Step 24.2.1.8

Combine $3$ and $\frac{1}{4}$ .

$f (\frac{7 π}{6}) = \frac{3}{4} + 3 \sin (\frac{7 π}{6})$

Step 24.2.1.9

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 third quadrant.

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

Step 24.2.1.10

The exact value of $\sin (\frac{π}{6})$ is $\frac{1}{2}$ .

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

Step 24.2.1.11

Multiply $3 (- \frac{1}{2})$ .

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Step 24.2.1.11.1

Multiply $- 1$ by $3$ .

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

Step 24.2.1.11.2

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

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

Step 24.2.1.12

Move the negative in front of the fraction.

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

Step 24.2.2

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

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

Step 24.2.3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 24.2.3.1

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

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

Step 24.2.3.2

Multiply $2$ by $2$ .

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

Step 24.2.4

Combine the numerators over the common denominator.

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

Step 24.2.5

Simplify the numerator.

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Step 24.2.5.1

Multiply $- 3$ by $2$ .

$f (\frac{7 π}{6}) = \frac{3 - 6}{4}$

Step 24.2.5.2

Subtract $6$ from $3$ .

$f (\frac{7 π}{6}) = \frac{- 3}{4}$

Step 24.2.6

Move the negative in front of the fraction.

$f (\frac{7 π}{6}) = - \frac{3}{4}$

Step 24.2.7

The final answer is $- \frac{3}{4}$ .

$y = - \frac{3}{4}$

Step 25

These are the local extrema for $f (x) = 3 \sin^{2} (x) + 3 \sin (x)$ .

$(\frac{π}{2}, 6)$ is a local maxima

$(\frac{3 π}{2}, 0)$ is a local maxima

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

$(\frac{7 π}{6}, - \frac{3}{4})$ is a local minima

Step 26