Calculus Examples

$9 \cos^{2} (x) - 18 \sin (x)$

Step 1

Write $9 \cos^{2} (x) - 18 \sin (x)$ as a function.

$f (x) = 9 \cos^{2} (x) - 18 \sin (x)$

Step 2

Find the second derivative.

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Step 2.1

Find the first derivative.

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Step 2.1.1

By the Sum Rule, the derivative of $9 \cos^{2} (x) - 18 \sin (x)$ with respect to $x$ is $\frac{d}{d x} [9 \cos^{2} (x)] + \frac{d}{d x} [- 18 \sin (x)]$ .

$\frac{d}{d x} [9 \cos^{2} (x)] + \frac{d}{d x} [- 18 \sin (x)]$

Step 2.1.2

Evaluate $\frac{d}{d x} [9 \cos^{2} (x)]$ .

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Step 2.1.2.1

Since $9$ is constant with respect to $x$ , the derivative of $9 \cos^{2} (x)$ with respect to $x$ is $9 \frac{d}{d x} [\cos^{2} (x)]$ .

$9 \frac{d}{d x} [\cos^{2} (x)] + \frac{d}{d x} [- 18 \sin (x)]$

Step 2.1.2.2

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 $g (x) = \cos (x)$ .

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Step 2.1.2.2.1

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

$9 (\frac{d}{d u} [u^{2}] \frac{d}{d x} [\cos (x)]) + \frac{d}{d x} [- 18 \sin (x)]$

Step 2.1.2.2.2

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

$9 (2 u \frac{d}{d x} [\cos (x)]) + \frac{d}{d x} [- 18 \sin (x)]$

Step 2.1.2.2.3

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

$9 (2 \cos (x) \frac{d}{d x} [\cos (x)]) + \frac{d}{d x} [- 18 \sin (x)]$

Step 2.1.2.3

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

$9 (2 \cos (x) (- \sin (x))) + \frac{d}{d x} [- 18 \sin (x)]$

Step 2.1.2.4

Multiply $- 1$ by $2$ .

$9 (- 2 \cos (x) \sin (x)) + \frac{d}{d x} [- 18 \sin (x)]$

Step 2.1.2.5

Multiply $- 2$ by $9$ .

$- 18 \cos (x) \sin (x) + \frac{d}{d x} [- 18 \sin (x)]$

Step 2.1.3

Evaluate $\frac{d}{d x} [- 18 \sin (x)]$ .

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Step 2.1.3.1

Since $- 18$ is constant with respect to $x$ , the derivative of $- 18 \sin (x)$ with respect to $x$ is $- 18 \frac{d}{d x} [\sin (x)]$ .

$- 18 \cos (x) \sin (x) - 18 \frac{d}{d x} [\sin (x)]$

Step 2.1.3.2

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

$f' (x) = - 18 \cos (x) \sin (x) - 18 \cos (x)$

Step 2.2

Find the second derivative.

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Step 2.2.1

By the Sum Rule, the derivative of $- 18 \cos (x) \sin (x) - 18 \cos (x)$ with respect to $x$ is $\frac{d}{d x} [- 18 \cos (x) \sin (x)] + \frac{d}{d x} [- 18 \cos (x)]$ .

$\frac{d}{d x} [- 18 \cos (x) \sin (x)] + \frac{d}{d x} [- 18 \cos (x)]$

Step 2.2.2

Evaluate $\frac{d}{d x} [- 18 \cos (x) \sin (x)]$ .

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Step 2.2.2.1

Since $- 18$ is constant with respect to $x$ , the derivative of $- 18 \cos (x) \sin (x)$ with respect to $x$ is $- 18 \frac{d}{d x} [\cos (x) \sin (x)]$ .

$- 18 \frac{d}{d x} [\cos (x) \sin (x)] + \frac{d}{d x} [- 18 \cos (x)]$

Step 2.2.2.2

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 $g (x) = \sin (x)$ .

$- 18 (\cos (x) \frac{d}{d x} [\sin (x)] + \sin (x) \frac{d}{d x} [\cos (x)]) + \frac{d}{d x} [- 18 \cos (x)]$

Step 2.2.2.3

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

$- 18 (\cos (x) \cos (x) + \sin (x) \frac{d}{d x} [\cos (x)]) + \frac{d}{d x} [- 18 \cos (x)]$

Step 2.2.2.4

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

$- 18 (\cos (x) \cos (x) + \sin (x) (- \sin (x))) + \frac{d}{d x} [- 18 \cos (x)]$

Step 2.2.2.5

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

$- 18 (\cos^{1} (x) \cos (x) + \sin (x) (- \sin (x))) + \frac{d}{d x} [- 18 \cos (x)]$

Step 2.2.2.6

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

$- 18 (\cos^{1} (x) \cos^{1} (x) + \sin (x) (- \sin (x))) + \frac{d}{d x} [- 18 \cos (x)]$

Step 2.2.2.7

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

$- 18 ({\cos (x)}^{1 + 1} + \sin (x) (- \sin (x))) + \frac{d}{d x} [- 18 \cos (x)]$

Step 2.2.2.8

Add $1$ and $1$ .

$- 18 (\cos^{2} (x) + \sin (x) (- \sin (x))) + \frac{d}{d x} [- 18 \cos (x)]$

Step 2.2.2.9

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

$- 18 (\cos^{2} (x) - (\sin^{1} (x) \sin (x))) + \frac{d}{d x} [- 18 \cos (x)]$

Step 2.2.2.10

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

$- 18 (\cos^{2} (x) - (\sin^{1} (x) \sin^{1} (x))) + \frac{d}{d x} [- 18 \cos (x)]$

Step 2.2.2.11

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

$- 18 (\cos^{2} (x) - {\sin (x)}^{1 + 1}) + \frac{d}{d x} [- 18 \cos (x)]$

Step 2.2.2.12

Add $1$ and $1$ .

$- 18 (\cos^{2} (x) - \sin^{2} (x)) + \frac{d}{d x} [- 18 \cos (x)]$

Step 2.2.3

Evaluate $\frac{d}{d x} [- 18 \cos (x)]$ .

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Step 2.2.3.1

Since $- 18$ is constant with respect to $x$ , the derivative of $- 18 \cos (x)$ with respect to $x$ is $- 18 \frac{d}{d x} [\cos (x)]$ .

$- 18 (\cos^{2} (x) - \sin^{2} (x)) - 18 \frac{d}{d x} [\cos (x)]$

Step 2.2.3.2

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

$- 18 (\cos^{2} (x) - \sin^{2} (x)) - 18 (- \sin (x))$

Step 2.2.3.3

Multiply $- 1$ by $- 18$ .

$- 18 (\cos^{2} (x) - \sin^{2} (x)) + 18 \sin (x)$

Step 2.2.4

Simplify.

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Step 2.2.4.1

Apply the distributive property.

$- 18 \cos^{2} (x) - 18 (- \sin^{2} (x)) + 18 \sin (x)$

Step 2.2.4.2

Multiply $- 1$ by $- 18$ .

$f'' (x) = - 18 \cos^{2} (x) + 18 \sin^{2} (x) + 18 \sin (x)$

Step 2.3

The second derivative of $f (x)$ with respect to $x$ is $- 18 \cos^{2} (x) + 18 \sin^{2} (x) + 18 \sin (x)$ .

$- 18 \cos^{2} (x) + 18 \sin^{2} (x) + 18 \sin (x)$

Step 3

Set the second derivative equal to $0$ then solve the equation $- 18 \cos^{2} (x) + 18 \sin^{2} (x) + 18 \sin (x) = 0$ .

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Step 3.1

Set the second derivative equal to $0$ .

$- 18 \cos^{2} (x) + 18 \sin^{2} (x) + 18 \sin (x) = 0$

Step 3.2

Replace the $- 18 \cos^{2} (x)$ with $- 18 (1 - \sin^{2} (x))$ based on the $\sin^{2} (x) + \cos^{2} (x) = 1$ identity.

$- 18 (1 - \sin^{2} (x)) + 18 \sin^{2} (x) + 18 \sin (x) = 0$

Step 3.3

Simplify each term.

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Step 3.3.1

Apply the distributive property.

$- 18 \cdot 1 - 18 (- \sin^{2} (x)) + 18 \sin^{2} (x) + 18 \sin (x) = 0$

Step 3.3.2

Multiply $- 18$ by $1$ .

$- 18 - 18 (- \sin^{2} (x)) + 18 \sin^{2} (x) + 18 \sin (x) = 0$

Step 3.3.3

Multiply $- 1$ by $- 18$ .

$- 18 + 18 \sin^{2} (x) + 18 \sin^{2} (x) + 18 \sin (x) = 0$

Step 3.4

Add $18 \sin^{2} (x)$ and $18 \sin^{2} (x)$ .

$- 18 + 36 \sin^{2} (x) + 18 \sin (x) = 0$

Step 3.5

Reorder the polynomial.

$36 \sin^{2} (x) + 18 \sin (x) - 18 = 0$

Step 3.6

Substitute $u$ for $\sin (x)$ .

$36 {(u)}^{2} + 18 (u) - 18 = 0$

Step 3.7

Factor the left side of the equation.

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Step 3.7.1

Factor $18$ out of $36 u^{2} + 18 u - 18$ .

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Step 3.7.1.1

Factor $18$ out of $36 u^{2}$ .

$18 (2 u^{2}) + 18 u - 18 = 0$

Step 3.7.1.2

Factor $18$ out of $18 u$ .

$18 (2 u^{2}) + 18 u - 18 = 0$

Step 3.7.1.3

Factor $18$ out of $- 18$ .

$18 (2 u^{2}) + 18 u + 18 (- 1) = 0$

Step 3.7.1.4

Factor $18$ out of $18 (2 u^{2}) + 18 u$ .

$18 (2 u^{2} + u) + 18 (- 1) = 0$

Step 3.7.1.5

Factor $18$ out of $18 (2 u^{2} + u) + 18 (- 1)$ .

$18 (2 u^{2} + u - 1) = 0$

Step 3.7.2

Factor.

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Step 3.7.2.1

Factor by grouping.

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Step 3.7.2.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 1 = - 2$ and whose sum is $b = 1$ .

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Step 3.7.2.1.1.1

Multiply by $1$ .

$18 (2 u^{2} + 1 u - 1) = 0$

Step 3.7.2.1.1.2

Rewrite $1$ as $- 1$ plus $2$

$18 (2 u^{2} + (- 1 + 2) u - 1) = 0$

Step 3.7.2.1.1.3

Apply the distributive property.

$18 (2 u^{2} - 1 u + 2 u - 1) = 0$

Step 3.7.2.1.2

Factor out the greatest common factor from each group.

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Step 3.7.2.1.2.1

Group the first two terms and the last two terms.

$18 (2 u^{2} - 1 u + 2 u - 1) = 0$

Step 3.7.2.1.2.2

Factor out the greatest common factor (GCF) from each group.

$18 (u (2 u - 1) + 1 (2 u - 1)) = 0$

Step 3.7.2.1.3

Factor the polynomial by factoring out the greatest common factor, $2 u - 1$ .

$18 ((2 u - 1) (u + 1)) = 0$

Step 3.7.2.2

Remove unnecessary parentheses.

$18 (2 u - 1) (u + 1) = 0$

Step 3.8

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

$2 u - 1 = 0$

$u + 1 = 0$

Step 3.9

Set $2 u - 1$ equal to $0$ and solve for $u$ .

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Step 3.9.1

Set $2 u - 1$ equal to $0$ .

$2 u - 1 = 0$

Step 3.9.2

Solve $2 u - 1 = 0$ for $u$ .

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Step 3.9.2.1

Add $1$ to both sides of the equation.

$2 u = 1$

Step 3.9.2.2

Divide each term in $2 u = 1$ by $2$ and simplify.

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Step 3.9.2.2.1

Divide each term in $2 u = 1$ by $2$ .

$\frac{2 u}{2} = \frac{1}{2}$

Step 3.9.2.2.2

Simplify the left side.

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Step 3.9.2.2.2.1

Cancel the common factor of $2$ .

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Step 3.9.2.2.2.1.1

Cancel the common factor.

$\frac{2 u}{2} = \frac{1}{2}$

Step 3.9.2.2.2.1.2

Divide $u$ by $1$ .

$u = \frac{1}{2}$

Step 3.10

Set $u + 1$ equal to $0$ and solve for $u$ .

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Step 3.10.1

Set $u + 1$ equal to $0$ .

$u + 1 = 0$

Step 3.10.2

Subtract $1$ from both sides of the equation.

$u = - 1$

Step 3.11

The final solution is all the values that make $18 (2 u - 1) (u + 1) = 0$ true.

$u = \frac{1}{2}, - 1$

Step 3.12

Substitute $\sin (x)$ for $u$ .

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

Step 3.13

Set up each of the solutions to solve for $x$ .

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

$\sin (x) = - 1$

Step 3.14

Solve for $x$ in $\sin (x) = \frac{1}{2}$ .

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Step 3.14.1

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

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

Step 3.14.2

Simplify the right side.

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Step 3.14.2.1

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

$x = \frac{π}{6}$

Step 3.14.3

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.

$x = π - \frac{π}{6}$

Step 3.14.4

Simplify $π - \frac{π}{6}$ .

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Step 3.14.4.1

To write $π$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

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

Step 3.14.4.2

Combine fractions.

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Step 3.14.4.2.1

Combine $π$ and $\frac{6}{6}$ .

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

Step 3.14.4.2.2

Combine the numerators over the common denominator.

$x = \frac{π \cdot 6 - π}{6}$

Step 3.14.4.3

Simplify the numerator.

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Step 3.14.4.3.1

Move $6$ to the left of $π$ .

$x = \frac{6 \cdot π - π}{6}$

Step 3.14.4.3.2

Subtract $π$ from $6 π$ .

$x = \frac{5 π}{6}$

Step 3.14.5

Find the period of $\sin (x)$ .

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Step 3.14.5.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 3.14.5.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 3.14.5.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 3.14.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 3.14.6

The period of the $\sin (x)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = \frac{π}{6} + 2 π n, \frac{5 π}{6} + 2 π n$ , for any integer $n$

Step 3.15

Solve for $x$ in $\sin (x) = - 1$ .

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Step 3.15.1

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

$x = \arcsin (- 1)$

Step 3.15.2

Simplify the right side.

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Step 3.15.2.1

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

$x = - \frac{π}{2}$

Step 3.15.3

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.

$x = 2 π + \frac{π}{2} + π$

Step 3.15.4

Simplify the expression to find the second solution.

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Step 3.15.4.1

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

$x = 2 π + \frac{π}{2} + π - 2 π$

Step 3.15.4.2

The resulting angle of $\frac{3 π}{2}$ is positive, less than $2 π$ , and coterminal with $2 π + \frac{π}{2} + π$ .

$x = \frac{3 π}{2}$

Step 3.15.5

Find the period of $\sin (x)$ .

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Step 3.15.5.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 3.15.5.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 3.15.5.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 3.15.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 3.15.6

Add $2 π$ to every negative angle to get positive angles.

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Step 3.15.6.1

Add $2 π$ to $- \frac{π}{2}$ to find the positive angle.

$- \frac{π}{2} + 2 π$

Step 3.15.6.2

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

$2 π \cdot \frac{2}{2} - \frac{π}{2}$

Step 3.15.6.3

Combine fractions.

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Step 3.15.6.3.1

Combine $2 π$ and $\frac{2}{2}$ .

$\frac{2 π \cdot 2}{2} - \frac{π}{2}$

Step 3.15.6.3.2

Combine the numerators over the common denominator.

$\frac{2 π \cdot 2 - π}{2}$

Step 3.15.6.4

Simplify the numerator.

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Step 3.15.6.4.1

Multiply $2$ by $2$ .

$\frac{4 π - π}{2}$

Step 3.15.6.4.2

Subtract $π$ from $4 π$ .

$\frac{3 π}{2}$

Step 3.15.6.5

List the new angles.

$x = \frac{3 π}{2}$

Step 3.15.7

The period of the $\sin (x)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = \frac{3 π}{2} + 2 π n, \frac{3 π}{2} + 2 π n$ , for any integer $n$

Step 3.16

List all of the solutions.

$x = \frac{π}{6} + 2 π n, \frac{5 π}{6} + 2 π n, \frac{3 π}{2} + 2 π n, \frac{3 π}{2} + 2 π n$ , for any integer $n$

Step 3.17

Consolidate the answers.

$x = \frac{π}{6} + \frac{2 π n}{3}$ , for any integer $n$

Step 4

Find the points where the second derivative is $0$ .

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Step 4.1

Substitute $\frac{π}{6}$ in $f (x) = 9 \cos^{2} (x) - 18 \sin (x)$ to find the value of $y$ .

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Step 4.1.1

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

$f (\frac{π}{6}) = 9 \cos^{2} (\frac{π}{6}) - 18 \sin (\frac{π}{6})$

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Simplify each term.

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Step 4.1.2.1.1

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

$f (\frac{π}{6}) = 9 {(\frac{\sqrt{3}}{2})}^{2} - 18 \sin (\frac{π}{6})$

Step 4.1.2.1.2

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

$f (\frac{π}{6}) = 9 (\frac{{\sqrt{3}}^{2}}{2^{2}}) - 18 \sin (\frac{π}{6})$

Step 4.1.2.1.3

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

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Step 4.1.2.1.3.1

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

$f (\frac{π}{6}) = 9 (\frac{{(3^{\frac{1}{2}})}^{2}}{2^{2}}) - 18 \sin (\frac{π}{6})$

Step 4.1.2.1.3.2

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

$f (\frac{π}{6}) = 9 (\frac{3^{\frac{1}{2} \cdot 2}}{2^{2}}) - 18 \sin (\frac{π}{6})$

Step 4.1.2.1.3.3

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

$f (\frac{π}{6}) = 9 (\frac{3^{\frac{2}{2}}}{2^{2}}) - 18 \sin (\frac{π}{6})$

Step 4.1.2.1.3.4

Cancel the common factor of $2$ .

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Step 4.1.2.1.3.4.1

Cancel the common factor.

$f (\frac{π}{6}) = 9 (\frac{3^{\frac{2}{2}}}{2^{2}}) - 18 \sin (\frac{π}{6})$

Step 4.1.2.1.3.4.2

Rewrite the expression.

$f (\frac{π}{6}) = 9 (\frac{3}{2^{2}}) - 18 \sin (\frac{π}{6})$

Step 4.1.2.1.3.5

Evaluate the exponent.

$f (\frac{π}{6}) = 9 (\frac{3}{2^{2}}) - 18 \sin (\frac{π}{6})$

Step 4.1.2.1.4

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

$f (\frac{π}{6}) = 9 (\frac{3}{4}) - 18 \sin (\frac{π}{6})$

Step 4.1.2.1.5

Multiply $9 (\frac{3}{4})$ .

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Step 4.1.2.1.5.1

Combine $9$ and $\frac{3}{4}$ .

$f (\frac{π}{6}) = \frac{9 \cdot 3}{4} - 18 \sin (\frac{π}{6})$

Step 4.1.2.1.5.2

Multiply $9$ by $3$ .

$f (\frac{π}{6}) = \frac{27}{4} - 18 \sin (\frac{π}{6})$

Step 4.1.2.1.6

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

$f (\frac{π}{6}) = \frac{27}{4} - 18 (\frac{1}{2})$

Step 4.1.2.1.7

Cancel the common factor of $2$ .

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Step 4.1.2.1.7.1

Factor $2$ out of $- 18$ .

$f (\frac{π}{6}) = \frac{27}{4} + 2 (- 9) (\frac{1}{2})$

Step 4.1.2.1.7.2

Cancel the common factor.

$f (\frac{π}{6}) = \frac{27}{4} + 2 \cdot (- 9 (\frac{1}{2}))$

Step 4.1.2.1.7.3

Rewrite the expression.

$f (\frac{π}{6}) = \frac{27}{4} - 9$

Step 4.1.2.2

To write $- 9$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$f (\frac{π}{6}) = \frac{27}{4} - 9 \cdot \frac{4}{4}$

Step 4.1.2.3

Combine $- 9$ and $\frac{4}{4}$ .

$f (\frac{π}{6}) = \frac{27}{4} + \frac{- 9 \cdot 4}{4}$

Step 4.1.2.4

Combine the numerators over the common denominator.

$f (\frac{π}{6}) = \frac{27 - 9 \cdot 4}{4}$

Step 4.1.2.5

Simplify the numerator.

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Step 4.1.2.5.1

Multiply $- 9$ by $4$ .

$f (\frac{π}{6}) = \frac{27 - 36}{4}$

Step 4.1.2.5.2

Subtract $36$ from $27$ .

$f (\frac{π}{6}) = \frac{- 9}{4}$

Step 4.1.2.6

Move the negative in front of the fraction.

$f (\frac{π}{6}) = - \frac{9}{4}$

Step 4.1.2.7

The final answer is $- \frac{9}{4}$ .

$- \frac{9}{4}$

Step 4.2

The point found by substituting $\frac{π}{6}$ in $f (x) = 9 \cos^{2} (x) - 18 \sin (x)$ is $(\frac{π}{6}, - \frac{9}{4})$ . This point can be an inflection point.

$(\frac{π}{6}, - \frac{9}{4})$

Step 5

Split $(- \infty, \infty)$ into intervals around the points that could potentially be inflection points.

$(- \infty, \frac{π}{6}) \cup (\frac{π}{6}, \infty)$

Step 6

Substitute a value from the interval $(- \infty, \frac{π}{6})$ into the second derivative to determine if it is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $0.42359877$ in the expression.

$f'' (0.42359877) = - 18 \cos^{2} (0.42359877) + 18 \sin^{2} (0.42359877) + 18 \sin (0.42359877)$

Step 6.2

The final answer is $- 18 \cos^{2} (0.42359877) + 18 \sin^{2} (0.42359877) + 18 \sin (0.42359877)$ .

$- 18 \cos^{2} (0.42359877) + 18 \sin^{2} (0.42359877) + 18 \sin (0.42359877)$

Step 6.3

At $0.42359877$ , the second derivative is $- 4.51875903$ . Since this is negative, the second derivative is decreasing on the interval $(- \infty, \frac{π}{6})$

Decreasing on $(- \infty, \frac{π}{6})$ since $f'' (x) < 0$

Step 7

Substitute a value from the interval $(\frac{π}{6}, \infty)$ into the second derivative to determine if it is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $0.62359877$ in the expression.

$f'' (0.62359877) = - 18 \cos^{2} (0.62359877) + 18 \sin^{2} (0.62359877) + 18 \sin (0.62359877)$

Step 7.2

The final answer is $- 18 \cos^{2} (0.62359877) + 18 \sin^{2} (0.62359877) + 18 \sin (0.62359877)$ .

$- 18 \cos^{2} (0.62359877) + 18 \sin^{2} (0.62359877) + 18 \sin (0.62359877)$

Step 7.3

At $0.62359877$ , the second derivative is $4.7876356$ . Since this is positive, the second derivative is increasing on the interval $(\frac{π}{6}, \infty)$ .

Increasing on $(\frac{π}{6}, \infty)$ since $f'' (x) > 0$

Step 8

An inflection point is a point on a curve at which the concavity changes sign from plus to minus or from minus to plus. The inflection point in this case is $(\frac{π}{6}, - \frac{9}{4})$ .

$(\frac{π}{6}, - \frac{9}{4})$

Step 9