Calculus Examples

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

Step 1

Find the second derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

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

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

Step 1.1.2

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

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Step 1.1.2.1

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

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

Step 1.1.2.2

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

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

Step 1.1.2.3

Multiply $2$ by $3$ .

$6 x + \frac{d}{d x} [- 3 \sin (2 x)]$

Step 1.1.3

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

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Step 1.1.3.1

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

$6 x - 3 \frac{d}{d x} [\sin (2 x)]$

Step 1.1.3.2

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

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Step 1.1.3.2.1

To apply the Chain Rule, set $u$ as $2 x$ .

$6 x - 3 (\frac{d}{d u} [\sin (u)] \frac{d}{d x} [2 x])$

Step 1.1.3.2.2

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

$6 x - 3 (\cos (u) \frac{d}{d x} [2 x])$

Step 1.1.3.2.3

Replace all occurrences of $u$ with $2 x$ .

$6 x - 3 (\cos (2 x) \frac{d}{d x} [2 x])$

Step 1.1.3.3

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

$6 x - 3 (\cos (2 x) (2 \frac{d}{d x} [x]))$

Step 1.1.3.4

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

$6 x - 3 (\cos (2 x) (2 \cdot 1))$

Step 1.1.3.5

Multiply $2$ by $1$ .

$6 x - 3 (\cos (2 x) \cdot 2)$

Step 1.1.3.6

Move $2$ to the left of $\cos (2 x)$ .

$6 x - 3 (2 \cdot \cos (2 x))$

Step 1.1.3.7

Multiply $2$ by $- 3$ .

$f' (x) = 6 x - 6 \cos (2 x)$

Step 1.2

Find the second derivative.

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Step 1.2.1

By the Sum Rule, the derivative of $6 x - 6 \cos (2 x)$ with respect to $x$ is $\frac{d}{d x} [6 x] + \frac{d}{d x} [- 6 \cos (2 x)]$ .

$\frac{d}{d x} [6 x] + \frac{d}{d x} [- 6 \cos (2 x)]$

Step 1.2.2

Evaluate $\frac{d}{d x} [6 x]$ .

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Step 1.2.2.1

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

$6 \frac{d}{d x} [x] + \frac{d}{d x} [- 6 \cos (2 x)]$

Step 1.2.2.2

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

$6 \cdot 1 + \frac{d}{d x} [- 6 \cos (2 x)]$

Step 1.2.2.3

Multiply $6$ by $1$ .

$6 + \frac{d}{d x} [- 6 \cos (2 x)]$

Step 1.2.3

Evaluate $\frac{d}{d x} [- 6 \cos (2 x)]$ .

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Step 1.2.3.1

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 \cos (2 x)$ with respect to $x$ is $- 6 \frac{d}{d x} [\cos (2 x)]$ .

$6 - 6 \frac{d}{d x} [\cos (2 x)]$

Step 1.2.3.2

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 $g (x) = 2 x$ .

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Step 1.2.3.2.1

To apply the Chain Rule, set $u$ as $2 x$ .

$6 - 6 (\frac{d}{d u} [\cos (u)] \frac{d}{d x} [2 x])$

Step 1.2.3.2.2

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

$6 - 6 (- \sin (u) \frac{d}{d x} [2 x])$

Step 1.2.3.2.3

Replace all occurrences of $u$ with $2 x$ .

$6 - 6 (- \sin (2 x) \frac{d}{d x} [2 x])$

Step 1.2.3.3

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

$6 - 6 (- \sin (2 x) (2 \frac{d}{d x} [x]))$

Step 1.2.3.4

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

$6 - 6 (- \sin (2 x) (2 \cdot 1))$

Step 1.2.3.5

Multiply $2$ by $1$ .

$6 - 6 (- \sin (2 x) \cdot 2)$

Step 1.2.3.6

Multiply $2$ by $- 1$ .

$6 - 6 (- 2 \sin (2 x))$

Step 1.2.3.7

Multiply $- 2$ by $- 6$ .

$f'' (x) = 6 + 12 \sin (2 x)$

Step 1.3

The second derivative of $f (x)$ with respect to $x$ is $6 + 12 \sin (2 x)$ .

$6 + 12 \sin (2 x)$

Step 2

Set the second derivative equal to $0$ then solve the equation $6 + 12 \sin (2 x) = 0$ .

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Step 2.1

Set the second derivative equal to $0$ .

$6 + 12 \sin (2 x) = 0$

Step 2.2

Subtract $6$ from both sides of the equation.

$12 \sin (2 x) = - 6$

Step 2.3

Divide each term in $12 \sin (2 x) = - 6$ by $12$ and simplify.

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Step 2.3.1

Divide each term in $12 \sin (2 x) = - 6$ by $12$ .

$\frac{12 \sin (2 x)}{12} = \frac{- 6}{12}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $12$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{12 \sin (2 x)}{12} = \frac{- 6}{12}$

Step 2.3.2.1.2

Divide $\sin (2 x)$ by $1$ .

$\sin (2 x) = \frac{- 6}{12}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Cancel the common factor of $- 6$ and $12$ .

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Step 2.3.3.1.1

Factor $6$ out of $- 6$ .

$\sin (2 x) = \frac{6 (- 1)}{12}$

Step 2.3.3.1.2

Cancel the common factors.

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Step 2.3.3.1.2.1

Factor $6$ out of $12$ .

$\sin (2 x) = \frac{6 \cdot - 1}{6 \cdot 2}$

Step 2.3.3.1.2.2

Cancel the common factor.

$\sin (2 x) = \frac{6 \cdot - 1}{6 \cdot 2}$

Step 2.3.3.1.2.3

Rewrite the expression.

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

Step 2.3.3.2

Move the negative in front of the fraction.

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

Step 2.4

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

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

Step 2.5

Simplify the right side.

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Step 2.5.1

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

$2 x = - \frac{π}{6}$

Step 2.6

Divide each term in $2 x = - \frac{π}{6}$ by $2$ and simplify.

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Step 2.6.1

Divide each term in $2 x = - \frac{π}{6}$ by $2$ .

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

Step 2.6.2

Simplify the left side.

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Step 2.6.2.1

Cancel the common factor of $2$ .

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Step 2.6.2.1.1

Cancel the common factor.

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

Step 2.6.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \frac{π}{6}}{2}$

Step 2.6.3

Simplify the right side.

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Step 2.6.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{π}{6} \cdot \frac{1}{2}$

Step 2.6.3.2

Multiply $- \frac{π}{6} \cdot \frac{1}{2}$ .

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Step 2.6.3.2.1

Multiply $\frac{1}{2}$ by $\frac{π}{6}$ .

$x = - \frac{π}{2 \cdot 6}$

Step 2.6.3.2.2

Multiply $2$ by $6$ .

$x = - \frac{π}{12}$

Step 2.7

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.

$2 x = 2 π + \frac{π}{6} + π$

Step 2.8

Simplify the expression to find the second solution.

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Step 2.8.1

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

$2 x = 2 π + \frac{π}{6} + π - 2 π$

Step 2.8.2

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

$2 x = \frac{7 π}{6}$

Step 2.8.3

Divide each term in $2 x = \frac{7 π}{6}$ by $2$ and simplify.

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Step 2.8.3.1

Divide each term in $2 x = \frac{7 π}{6}$ by $2$ .

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

Step 2.8.3.2

Simplify the left side.

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Step 2.8.3.2.1

Cancel the common factor of $2$ .

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Step 2.8.3.2.1.1

Cancel the common factor.

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

Step 2.8.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{7 π}{6}}{2}$

Step 2.8.3.3

Simplify the right side.

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Step 2.8.3.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{7 π}{6} \cdot \frac{1}{2}$

Step 2.8.3.3.2

Multiply $\frac{7 π}{6} \cdot \frac{1}{2}$ .

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Step 2.8.3.3.2.1

Multiply $\frac{7 π}{6}$ by $\frac{1}{2}$ .

$x = \frac{7 π}{6 \cdot 2}$

Step 2.8.3.3.2.2

Multiply $6$ by $2$ .

$x = \frac{7 π}{12}$

Step 2.9

Find the period of $\sin (2 x)$ .

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Step 2.9.1

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

$\frac{2 π}{| b |}$

Step 2.9.2

Replace $b$ with $2$ in the formula for period.

$\frac{2 π}{| 2 |}$

Step 2.9.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{2 π}{2}$

Step 2.9.4

Cancel the common factor of $2$ .

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Step 2.9.4.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 2.9.4.2

Divide $π$ by $1$ .

$π$

Step 2.10

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

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Step 2.10.1

Add $π$ to $- \frac{π}{12}$ to find the positive angle.

$- \frac{π}{12} + π$

Step 2.10.2

To write $π$ as a fraction with a common denominator, multiply by $\frac{12}{12}$ .

$π \cdot \frac{12}{12} - \frac{π}{12}$

Step 2.10.3

Combine fractions.

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Step 2.10.3.1

Combine $π$ and $\frac{12}{12}$ .

$\frac{π \cdot 12}{12} - \frac{π}{12}$

Step 2.10.3.2

Combine the numerators over the common denominator.

$\frac{π \cdot 12 - π}{12}$

Step 2.10.4

Simplify the numerator.

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Step 2.10.4.1

Move $12$ to the left of $π$ .

$\frac{12 \cdot π - π}{12}$

Step 2.10.4.2

Subtract $π$ from $12 π$ .

$\frac{11 π}{12}$

Step 2.10.5

List the new angles.

$x = \frac{11 π}{12}$

Step 2.11

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

$x = \frac{7 π}{12} + π n, \frac{11 π}{12} + π n$ , for any integer $n$

Step 3

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

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Step 3.1

Substitute $\frac{7 π}{12}$ in $f (x) = 3 x^{2} - 3 \sin (2 x)$ to find the value of $y$ .

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Step 3.1.1

Replace the variable $x$ with $\frac{7 π}{12}$ in the expression.

$f (\frac{7 π}{12}) = 3 {(\frac{7 π}{12})}^{2} - 3 \sin (2 (\frac{7 π}{12}))$

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Simplify each term.

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Step 3.1.2.1.1

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

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Step 3.1.2.1.1.1

Apply the product rule to $\frac{7 π}{12}$ .

$f (\frac{7 π}{12}) = 3 (\frac{{(7 π)}^{2}}{12^{2}}) - 3 \sin (2 (\frac{7 π}{12}))$

Step 3.1.2.1.1.2

Apply the product rule to $7 π$ .

$f (\frac{7 π}{12}) = 3 (\frac{7^{2} π^{2}}{12^{2}}) - 3 \sin (2 (\frac{7 π}{12}))$

Step 3.1.2.1.2

Raise $7$ to the power of $2$ .

$f (\frac{7 π}{12}) = 3 (\frac{49 π^{2}}{12^{2}}) - 3 \sin (2 (\frac{7 π}{12}))$

Step 3.1.2.1.3

Raise $12$ to the power of $2$ .

$f (\frac{7 π}{12}) = 3 (\frac{49 π^{2}}{144}) - 3 \sin (2 (\frac{7 π}{12}))$

Step 3.1.2.1.4

Cancel the common factor of $3$ .

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Step 3.1.2.1.4.1

Factor $3$ out of $144$ .

$f (\frac{7 π}{12}) = 3 (\frac{49 π^{2}}{3 (48)}) - 3 \sin (2 (\frac{7 π}{12}))$

Step 3.1.2.1.4.2

Cancel the common factor.

$f (\frac{7 π}{12}) = 3 (\frac{49 π^{2}}{3 \cdot 48}) - 3 \sin (2 (\frac{7 π}{12}))$

Step 3.1.2.1.4.3

Rewrite the expression.

$f (\frac{7 π}{12}) = \frac{49 π^{2}}{48} - 3 \sin (2 (\frac{7 π}{12}))$

Step 3.1.2.1.5

Cancel the common factor of $2$ .

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Step 3.1.2.1.5.1

Factor $2$ out of $12$ .

$f (\frac{7 π}{12}) = \frac{49 π^{2}}{48} - 3 \sin (2 (\frac{7 π}{2 (6)}))$

Step 3.1.2.1.5.2

Cancel the common factor.

$f (\frac{7 π}{12}) = \frac{49 π^{2}}{48} - 3 \sin (2 (\frac{7 π}{2 \cdot 6}))$

Step 3.1.2.1.5.3

Rewrite the expression.

$f (\frac{7 π}{12}) = \frac{49 π^{2}}{48} - 3 \sin (\frac{7 π}{6})$

Step 3.1.2.1.6

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 third quadrant.

$f (\frac{7 π}{12}) = \frac{49 π^{2}}{48} - 3 (- \sin (\frac{π}{6}))$

Step 3.1.2.1.7

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

$f (\frac{7 π}{12}) = \frac{49 π^{2}}{48} - 3 (- \frac{1}{2})$

Step 3.1.2.1.8

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

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Step 3.1.2.1.8.1

Multiply $- 1$ by $- 3$ .

$f (\frac{7 π}{12}) = \frac{49 π^{2}}{48} + 3 (\frac{1}{2})$

Step 3.1.2.1.8.2

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

$f (\frac{7 π}{12}) = \frac{49 π^{2}}{48} + \frac{3}{2}$

Step 3.1.2.2

The final answer is $\frac{49 π^{2}}{48} + \frac{3}{2}$ .

$\frac{49 π^{2}}{48} + \frac{3}{2}$

Step 3.2

The point found by substituting $\frac{7 π}{12}$ in $f (x) = 3 x^{2} - 3 \sin (2 x)$ is $(\frac{7 π}{12}, \frac{49 π^{2}}{48} + \frac{3}{2})$ . This point can be an inflection point.

$(\frac{7 π}{12}, \frac{49 π^{2}}{48} + \frac{3}{2})$

Step 3.3

Substitute $\frac{11 π}{12}$ in $f (x) = 3 x^{2} - 3 \sin (2 x)$ to find the value of $y$ .

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Step 3.3.1

Replace the variable $x$ with $\frac{11 π}{12}$ in the expression.

$f (\frac{11 π}{12}) = 3 {(\frac{11 π}{12})}^{2} - 3 \sin (2 (\frac{11 π}{12}))$

Step 3.3.2

Simplify the result.

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Step 3.3.2.1

Simplify each term.

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Step 3.3.2.1.1

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

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Step 3.3.2.1.1.1

Apply the product rule to $\frac{11 π}{12}$ .

$f (\frac{11 π}{12}) = 3 (\frac{{(11 π)}^{2}}{12^{2}}) - 3 \sin (2 (\frac{11 π}{12}))$

Step 3.3.2.1.1.2

Apply the product rule to $11 π$ .

$f (\frac{11 π}{12}) = 3 (\frac{11^{2} π^{2}}{12^{2}}) - 3 \sin (2 (\frac{11 π}{12}))$

Step 3.3.2.1.2

Raise $11$ to the power of $2$ .

$f (\frac{11 π}{12}) = 3 (\frac{121 π^{2}}{12^{2}}) - 3 \sin (2 (\frac{11 π}{12}))$

Step 3.3.2.1.3

Raise $12$ to the power of $2$ .

$f (\frac{11 π}{12}) = 3 (\frac{121 π^{2}}{144}) - 3 \sin (2 (\frac{11 π}{12}))$

Step 3.3.2.1.4

Cancel the common factor of $3$ .

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Step 3.3.2.1.4.1

Factor $3$ out of $144$ .

$f (\frac{11 π}{12}) = 3 (\frac{121 π^{2}}{3 (48)}) - 3 \sin (2 (\frac{11 π}{12}))$

Step 3.3.2.1.4.2

Cancel the common factor.

$f (\frac{11 π}{12}) = 3 (\frac{121 π^{2}}{3 \cdot 48}) - 3 \sin (2 (\frac{11 π}{12}))$

Step 3.3.2.1.4.3

Rewrite the expression.

$f (\frac{11 π}{12}) = \frac{121 π^{2}}{48} - 3 \sin (2 (\frac{11 π}{12}))$

Step 3.3.2.1.5

Cancel the common factor of $2$ .

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Step 3.3.2.1.5.1

Factor $2$ out of $12$ .

$f (\frac{11 π}{12}) = \frac{121 π^{2}}{48} - 3 \sin (2 (\frac{11 π}{2 (6)}))$

Step 3.3.2.1.5.2

Cancel the common factor.

$f (\frac{11 π}{12}) = \frac{121 π^{2}}{48} - 3 \sin (2 (\frac{11 π}{2 \cdot 6}))$

Step 3.3.2.1.5.3

Rewrite the expression.

$f (\frac{11 π}{12}) = \frac{121 π^{2}}{48} - 3 \sin (\frac{11 π}{6})$

Step 3.3.2.1.6

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.

$f (\frac{11 π}{12}) = \frac{121 π^{2}}{48} - 3 (- \sin (\frac{π}{6}))$

Step 3.3.2.1.7

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

$f (\frac{11 π}{12}) = \frac{121 π^{2}}{48} - 3 (- \frac{1}{2})$

Step 3.3.2.1.8

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

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Step 3.3.2.1.8.1

Multiply $- 1$ by $- 3$ .

$f (\frac{11 π}{12}) = \frac{121 π^{2}}{48} + 3 (\frac{1}{2})$

Step 3.3.2.1.8.2

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

$f (\frac{11 π}{12}) = \frac{121 π^{2}}{48} + \frac{3}{2}$

Step 3.3.2.2

The final answer is $\frac{121 π^{2}}{48} + \frac{3}{2}$ .

$\frac{121 π^{2}}{48} + \frac{3}{2}$

Step 3.4

The point found by substituting $\frac{11 π}{12}$ in $f (x) = 3 x^{2} - 3 \sin (2 x)$ is $(\frac{11 π}{12}, \frac{121 π^{2}}{48} + \frac{3}{2})$ . This point can be an inflection point.

$(\frac{11 π}{12}, \frac{121 π^{2}}{48} + \frac{3}{2})$

Step 3.5

Determine the points that could be inflection points.

$(\frac{7 π}{12}, \frac{49 π^{2}}{48} + \frac{3}{2}), (\frac{11 π}{12}, \frac{121 π^{2}}{48} + \frac{3}{2})$

Step 4

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

$(- \infty, \frac{7 π}{12}) \cup (\frac{7 π}{12}, \frac{11 π}{12}) \cup (\frac{11 π}{12}, \infty)$

Step 5

Substitute a value from the interval $(- \infty, \frac{7 π}{12})$ into the second derivative to determine if it is increasing or decreasing.

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Step 5.1

Replace the variable $x$ with $1.73259571$ in the expression.

$f'' (1.73259571) = 6 + 12 \sin (2 (1.73259571))$

Step 5.2

Simplify the result.

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Step 5.2.1

Multiply $2$ by $1.73259571$ .

$f'' (1.73259571) = 6 + 12 \sin (3.46519142)$

Step 5.2.2

The final answer is $6 + 12 \sin (3.46519142)$ .

$6 + 12 \sin (3.46519142)$

Step 5.3

At $1.73259571$ , the second derivative is $2.18423278$ . Since this is positive, the second derivative is increasing on the interval $(- \infty, \frac{7 π}{12})$ .

Increasing on $(- \infty, \frac{7 π}{12})$ since $f'' (x) > 0$

Step 6

Substitute a value from the interval $(\frac{7 π}{12}, \frac{11 π}{12})$ into the second derivative to determine if it is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $2.35619449$ in the expression.

$f'' (2.35619449) = 6 + 12 \sin (2 (2.35619449))$

Step 6.2

Simplify the result.

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Step 6.2.1

Multiply $2$ by $2.35619449$ .

$f'' (2.35619449) = 6 + 12 \sin (4.71238898)$

Step 6.2.2

The final answer is $6 + 12 \sin (4.71238898)$ .

$6 + 12 \sin (4.71238898)$

Step 6.3

At $2.35619449$ , the second derivative is $- 6$ . Since this is negative, the second derivative is decreasing on the interval $(\frac{7 π}{12}, \frac{11 π}{12})$

Decreasing on $(\frac{7 π}{12}, \frac{11 π}{12})$ since $f'' (x) < 0$

Step 7

Substitute a value from the interval $(\frac{11 π}{12}, \infty)$ into the second derivative to determine if it is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $2.97979326$ in the expression.

$f'' (2.97979326) = 6 + 12 \sin (2 (2.97979326))$

Step 7.2

Simplify the result.

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Step 7.2.1

Multiply $2$ by $2.97979326$ .

$f'' (2.97979326) = 6 + 12 \sin (5.95958653)$

Step 7.2.2

The final answer is $6 + 12 \sin (5.95958653)$ .

$6 + 12 \sin (5.95958653)$

Step 7.3

At $2.97979326$ , the second derivative is $2.18423278$ . Since this is positive, the second derivative is increasing on the interval $(\frac{11 π}{12}, \infty)$ .

Increasing on $(\frac{11 π}{12}, \infty)$ 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 points in this case are $(\frac{7 π}{12}, \frac{49 π^{2}}{48} + \frac{3}{2}), (\frac{11 π}{12}, \frac{121 π^{2}}{48} + \frac{3}{2})$ .

$(\frac{7 π}{12}, \frac{49 π^{2}}{48} + \frac{3}{2}), (\frac{11 π}{12}, \frac{121 π^{2}}{48} + \frac{3}{2})$

Step 9