Calculus Examples

$y = 4 \sin (\frac{4 π}{3} x)$

Step 1

Find the first derivative of the function.

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Step 1.1

Differentiate using the Constant Multiple Rule.

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Step 1.1.1

Combine $\frac{4 π}{3}$ and $x$ .

$\frac{d}{d x} [4 \sin (\frac{4 π x}{3})]$

Step 1.1.2

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

$4 \frac{d}{d x} [\sin (\frac{4 π x}{3})]$

Step 1.2

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

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Step 1.2.1

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

$4 (\frac{d}{d u} [\sin (u)] \frac{d}{d x} [\frac{4 π x}{3}])$

Step 1.2.2

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

$4 (\cos (u) \frac{d}{d x} [\frac{4 π x}{3}])$

Step 1.2.3

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

$4 (\cos (\frac{4 π x}{3}) \frac{d}{d x} [\frac{4 π x}{3}])$

Step 1.3

Differentiate.

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Step 1.3.1

Since $\frac{4 π}{3}$ is constant with respect to $x$ , the derivative of $\frac{4 π x}{3}$ with respect to $x$ is $\frac{4 π}{3} \frac{d}{d x} [x]$ .

$4 \cos (\frac{4 π x}{3}) (\frac{4 π}{3} \frac{d}{d x} [x])$

Step 1.3.2

Combine fractions.

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Step 1.3.2.1

Combine $\frac{4 π}{3}$ and $4$ .

$\frac{4 π \cdot 4}{3} \cos (\frac{4 π x}{3}) \frac{d}{d x} [x]$

Step 1.3.2.2

Multiply $4$ by $4$ .

$\frac{16 π}{3} \cos (\frac{4 π x}{3}) \frac{d}{d x} [x]$

Step 1.3.2.3

Combine $\frac{16 π}{3}$ and $\cos (\frac{4 π x}{3})$ .

$\frac{16 π \cos (\frac{4 π x}{3})}{3} \frac{d}{d x} [x]$

Step 1.3.3

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

$\frac{16 π \cos (\frac{4 π x}{3})}{3} \cdot 1$

Step 1.3.4

Multiply $\frac{16 π \cos (\frac{4 π x}{3})}{3}$ by $1$ .

$\frac{16 π \cos (\frac{4 π x}{3})}{3}$

Step 2

Find the second derivative of the function.

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Step 2.1

Since $\frac{16 π}{3}$ is constant with respect to $x$ , the derivative of $\frac{16 π \cos (\frac{4 π x}{3})}{3}$ with respect to $x$ is $\frac{16 π}{3} \frac{d}{d x} [\cos (\frac{4 π x}{3})]$ .

$f'' (x) = \frac{16 π}{3} \cdot \frac{d}{d x} (\cos (\frac{4 π x}{3}))$

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 $g (x) = \frac{4 π x}{3}$ .

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Step 2.2.1

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

$f'' (x) = \frac{16 π}{3} \cdot (\frac{d}{d u} (\cos (u)) \frac{d}{d x} (\frac{4 π x}{3}))$

Step 2.2.2

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

$f'' (x) = \frac{16 π}{3} \cdot (- \sin (u) \frac{d}{d x} \frac{4 π x}{3})$

Step 2.2.3

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

$f'' (x) = \frac{16 π}{3} \cdot (- \sin (\frac{4 π x}{3}) \frac{d}{d x} \frac{4 π x}{3})$

Step 2.3

Differentiate using the Constant Multiple Rule.

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Step 2.3.1

Combine $\frac{16 π}{3}$ and $\sin (\frac{4 π x}{3})$ .

$f'' (x) = - \frac{16 π \sin (\frac{4 π x}{3})}{3} \cdot \frac{d}{d x} \frac{4 π x}{3}$

Step 2.3.2

Since $\frac{4 π}{3}$ is constant with respect to $x$ , the derivative of $\frac{4 π x}{3}$ with respect to $x$ is $\frac{4 π}{3} \frac{d}{d x} [x]$ .

$f'' (x) = - \frac{16 π \sin (\frac{4 π x}{3})}{3} \cdot (\frac{4 π}{3} \cdot \frac{d}{d x} (x))$

Step 2.3.3

Combine fractions.

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Step 2.3.3.1

Multiply $\frac{4 π}{3}$ by $\frac{16 π \sin (\frac{4 π x}{3})}{3}$ .

$f'' (x) = - \frac{4 π (16 π \sin (\frac{4 π x}{3}))}{3 \cdot 3} \cdot \frac{d}{d x} x$

Step 2.3.3.2

Multiply $16$ by $4$ .

$f'' (x) = - \frac{64 π (π \sin (\frac{4 π x}{3}))}{3 \cdot 3} \cdot \frac{d}{d x} x$

Step 2.4

Raise $π$ to the power of $1$ .

$f'' (x) = - \frac{64 (π π) \sin (\frac{4 π x}{3})}{3 \cdot 3} \cdot \frac{d}{d x} x$

Step 2.5

Raise $π$ to the power of $1$ .

$f'' (x) = - \frac{64 (π π) \sin (\frac{4 π x}{3})}{3 \cdot 3} \cdot \frac{d}{d x} x$

Step 2.6

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

$f'' (x) = - \frac{64 π^{1 + 1} \sin (\frac{4 π x}{3})}{3 \cdot 3} \cdot \frac{d}{d x} x$

Step 2.7

Simplify the expression.

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Step 2.7.1

Add $1$ and $1$ .

$f'' (x) = - \frac{64 π^{2} \sin (\frac{4 π x}{3})}{3 \cdot 3} \cdot \frac{d}{d x} x$

Step 2.7.2

Multiply $3$ by $3$ .

$f'' (x) = - \frac{64 π^{2} \sin (\frac{4 π x}{3})}{9} \cdot \frac{d}{d x} x$

Step 2.8

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

$f'' (x) = - \frac{64 π^{2} \sin (\frac{4 π x}{3})}{9} \cdot 1$

Step 2.9

Multiply $- 1$ by $1$ .

$f'' (x) = - \frac{64 π^{2} \sin (\frac{4 π x}{3})}{9}$

Step 3

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

$\frac{16 π \cos (\frac{4 π x}{3})}{3} = 0$

Step 4

Set the numerator equal to zero.

$16 π \cos (\frac{4 π x}{3}) = 0$

Step 5

Solve the equation for $x$ .

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Step 5.1

Divide each term in $16 π \cos (\frac{4 π x}{3}) = 0$ by $16 π$ and simplify.

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Step 5.1.1

Divide each term in $16 π \cos (\frac{4 π x}{3}) = 0$ by $16 π$ .

$\frac{16 π \cos (\frac{4 π x}{3})}{16 π} = \frac{0}{16 π}$

Step 5.1.2

Simplify the left side.

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Step 5.1.2.1

Cancel the common factor of $16$ .

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Step 5.1.2.1.1

Cancel the common factor.

$\frac{16 π \cos (\frac{4 π x}{3})}{16 π} = \frac{0}{16 π}$

Step 5.1.2.1.2

Rewrite the expression.

$\frac{π \cos (\frac{4 π x}{3})}{π} = \frac{0}{16 π}$

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.

$\frac{π \cos (\frac{4 π x}{3})}{π} = \frac{0}{16 π}$

Step 5.1.2.2.2

Divide $\cos (\frac{4 π x}{3})$ by $1$ .

$\cos (\frac{4 π x}{3}) = \frac{0}{16 π}$

Step 5.1.3

Simplify the right side.

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Step 5.1.3.1

Cancel the common factor of $0$ and $16$ .

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Step 5.1.3.1.1

Factor $16$ out of $0$ .

$\cos (\frac{4 π x}{3}) = \frac{16 (0)}{16 π}$

Step 5.1.3.1.2

Cancel the common factors.

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Step 5.1.3.1.2.1

Factor $16$ out of $16 π$ .

$\cos (\frac{4 π x}{3}) = \frac{16 (0)}{16 (π)}$

Step 5.1.3.1.2.2

Cancel the common factor.

$\cos (\frac{4 π x}{3}) = \frac{16 \cdot 0}{16 π}$

Step 5.1.3.1.2.3

Rewrite the expression.

$\cos (\frac{4 π x}{3}) = \frac{0}{π}$

Step 5.1.3.2

Divide $0$ by $π$ .

$\cos (\frac{4 π x}{3}) = 0$

Step 5.2

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

$\frac{4 π x}{3} = \arccos (0)$

Step 5.3

Simplify the right side.

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Step 5.3.1

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

$\frac{4 π x}{3} = \frac{π}{2}$

Step 5.4

Multiply both sides of the equation by $\frac{3}{4 π}$ .

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

Step 5.5

Simplify both sides of the equation.

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Step 5.5.1

Simplify the left side.

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Step 5.5.1.1

Simplify $\frac{3}{4 π} \cdot \frac{4 π x}{3}$ .

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Step 5.5.1.1.1

Cancel the common factor of $3$ .

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Step 5.5.1.1.1.1

Cancel the common factor.

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

Step 5.5.1.1.1.2

Rewrite the expression.

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

Step 5.5.1.1.2

Cancel the common factor of $4 π$ .

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Step 5.5.1.1.2.1

Factor $4 π$ out of $4 π x$ .

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

Step 5.5.1.1.2.2

Cancel the common factor.

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

Step 5.5.1.1.2.3

Rewrite the expression.

$x = \frac{3}{4 π} \cdot \frac{π}{2}$

Step 5.5.2

Simplify the right side.

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Step 5.5.2.1

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

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Step 5.5.2.1.1

Cancel the common factor of $π$ .

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Step 5.5.2.1.1.1

Factor $π$ out of $4 π$ .

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

Step 5.5.2.1.1.2

Cancel the common factor.

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

Step 5.5.2.1.1.3

Rewrite the expression.

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

Step 5.5.2.1.2

Multiply $\frac{3}{4}$ by $\frac{1}{2}$ .

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

Step 5.5.2.1.3

Multiply $4$ by $2$ .

$x = \frac{3}{8}$

Step 5.6

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.

$\frac{4 π x}{3} = 2 π - \frac{π}{2}$

Step 5.7

Solve for $x$ .

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Step 5.7.1

Multiply both sides of the equation by $\frac{3}{4 π}$ .

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

Step 5.7.2

Simplify both sides of the equation.

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Step 5.7.2.1

Simplify the left side.

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Step 5.7.2.1.1

Simplify $\frac{3}{4 π} \cdot \frac{4 π x}{3}$ .

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Step 5.7.2.1.1.1

Cancel the common factor of $3$ .

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Step 5.7.2.1.1.1.1

Cancel the common factor.

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

Step 5.7.2.1.1.1.2

Rewrite the expression.

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

Step 5.7.2.1.1.2

Cancel the common factor of $4 π$ .

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Step 5.7.2.1.1.2.1

Factor $4 π$ out of $4 π x$ .

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

Step 5.7.2.1.1.2.2

Cancel the common factor.

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

Step 5.7.2.1.1.2.3

Rewrite the expression.

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

Step 5.7.2.2

Simplify the right side.

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Step 5.7.2.2.1

Simplify $\frac{3}{4 π} (2 π - \frac{π}{2})$ .

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Step 5.7.2.2.1.1

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

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

Step 5.7.2.2.1.2

Combine fractions.

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Step 5.7.2.2.1.2.1

Combine $2 π$ and $\frac{2}{2}$ .

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

Step 5.7.2.2.1.2.2

Combine the numerators over the common denominator.

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

Step 5.7.2.2.1.3

Simplify the numerator.

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Step 5.7.2.2.1.3.1

Multiply $2$ by $2$ .

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

Step 5.7.2.2.1.3.2

Subtract $π$ from $4 π$ .

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

Step 5.7.2.2.1.4

Simplify terms.

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Step 5.7.2.2.1.4.1

Cancel the common factor of $π$ .

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Step 5.7.2.2.1.4.1.1

Factor $π$ out of $4 π$ .

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

Step 5.7.2.2.1.4.1.2

Factor $π$ out of $3 π$ .

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

Step 5.7.2.2.1.4.1.3

Cancel the common factor.

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

Step 5.7.2.2.1.4.1.4

Rewrite the expression.

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

Step 5.7.2.2.1.4.2

Multiply $\frac{3}{4}$ by $\frac{3}{2}$ .

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

Step 5.7.2.2.1.4.3

Multiply.

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Step 5.7.2.2.1.4.3.1

Multiply $3$ by $3$ .

$x = \frac{9}{4 \cdot 2}$

Step 5.7.2.2.1.4.3.2

Multiply $4$ by $2$ .

$x = \frac{9}{8}$

Step 5.8

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

$x = \frac{3}{8}, \frac{9}{8}$

Step 6

Evaluate the second derivative at $x = \frac{3}{8}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- \frac{64 π^{2} \sin (\frac{4 π (\frac{3}{8})}{3})}{9}$

Step 7

Evaluate the second derivative.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Combine $\frac{3}{8}$ and $4$ .

$- \frac{64 π^{2} \sin (\frac{\frac{3 \cdot 4}{8} π}{3})}{9}$

Step 7.1.2

Combine $\frac{3 \cdot 4}{8}$ and $π$ .

$- \frac{64 π^{2} \sin (\frac{\frac{3 \cdot 4 π}{8}}{3})}{9}$

Step 7.2

Reduce the expression by cancelling the common factors.

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Step 7.2.1

Multiply $3$ by $4$ .

$- \frac{64 π^{2} \sin (\frac{\frac{12 π}{8}}{3})}{9}$

Step 7.2.2

Reduce the expression $\frac{12 π}{8}$ by cancelling the common factors.

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Step 7.2.2.1

Factor $4$ out of $12 π$ .

$- \frac{64 π^{2} \sin (\frac{\frac{4 (3 π)}{8}}{3})}{9}$

Step 7.2.2.2

Factor $4$ out of $8$ .

$- \frac{64 π^{2} \sin (\frac{\frac{4 (3 π)}{4 (2)}}{3})}{9}$

Step 7.2.2.3

Cancel the common factor.

$- \frac{64 π^{2} \sin (\frac{\frac{4 (3 π)}{4 \cdot 2}}{3})}{9}$

Step 7.2.2.4

Rewrite the expression.

$- \frac{64 π^{2} \sin (\frac{\frac{3 π}{2}}{3})}{9}$

Step 7.3

Simplify the numerator.

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Step 7.3.1

Multiply the numerator by the reciprocal of the denominator.

$- \frac{64 π^{2} \sin (\frac{3 π}{2} \cdot \frac{1}{3})}{9}$

Step 7.3.2

Cancel the common factor of $3$ .

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Step 7.3.2.1

Factor $3$ out of $3 π$ .

$- \frac{64 π^{2} \sin (\frac{3 (π)}{2} \cdot \frac{1}{3})}{9}$

Step 7.3.2.2

Cancel the common factor.

$- \frac{64 π^{2} \sin (\frac{3 π}{2} \cdot \frac{1}{3})}{9}$

Step 7.3.2.3

Rewrite the expression.

$- \frac{64 π^{2} \sin (\frac{π}{2})}{9}$

Step 7.3.3

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

$- \frac{64 π^{2} \cdot 1}{9}$

Step 7.3.4

Multiply $64$ by $1$ .

$- \frac{64 π^{2}}{9}$

Step 8

$x = \frac{3}{8}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = \frac{3}{8}$ is a local maximum

Step 9

Find the y-value when $x = \frac{3}{8}$ .

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Step 9.1

Replace the variable $x$ with $\frac{3}{8}$ in the expression.

$f (\frac{3}{8}) = 4 \sin (\frac{4 π}{3} \cdot (\frac{3}{8}))$

Step 9.2

Simplify the result.

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Step 9.2.1

Cancel the common factor of $4$ .

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Step 9.2.1.1

Factor $4$ out of $4 π$ .

$f (\frac{3}{8}) = 4 \sin (\frac{4 (π)}{3} \cdot \frac{3}{8})$

Step 9.2.1.2

Factor $4$ out of $8$ .

$f (\frac{3}{8}) = 4 \sin (\frac{4 (π)}{3} \cdot \frac{3}{4 (2)})$

Step 9.2.1.3

Cancel the common factor.

$f (\frac{3}{8}) = 4 \sin (\frac{4 π}{3} \cdot \frac{3}{4 \cdot 2})$

Step 9.2.1.4

Rewrite the expression.

$f (\frac{3}{8}) = 4 \sin (\frac{π}{3} \cdot \frac{3}{2})$

Step 9.2.2

Cancel the common factor of $3$ .

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Step 9.2.2.1

Cancel the common factor.

$f (\frac{3}{8}) = 4 \sin (\frac{π}{3} \cdot \frac{3}{2})$

Step 9.2.2.2

Rewrite the expression.

$f (\frac{3}{8}) = 4 \sin (π \cdot \frac{1}{2})$

Step 9.2.3

Combine $π$ and $\frac{1}{2}$ .

$f (\frac{3}{8}) = 4 \sin (\frac{π}{2})$

Step 9.2.4

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

$f (\frac{3}{8}) = 4 \cdot 1$

Step 9.2.5

Multiply $4$ by $1$ .

$f (\frac{3}{8}) = 4$

Step 9.2.6

The final answer is $4$ .

$y = 4$

Step 10

Evaluate the second derivative at $x = \frac{9}{8}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- \frac{64 π^{2} \sin (\frac{4 π (\frac{9}{8})}{3})}{9}$

Step 11

Evaluate the second derivative.

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Step 11.1

Simplify the numerator.

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Step 11.1.1

Combine $\frac{9}{8}$ and $4$ .

$- \frac{64 π^{2} \sin (\frac{\frac{9 \cdot 4}{8} π}{3})}{9}$

Step 11.1.2

Combine $\frac{9 \cdot 4}{8}$ and $π$ .

$- \frac{64 π^{2} \sin (\frac{\frac{9 \cdot 4 π}{8}}{3})}{9}$

Step 11.2

Reduce the expression by cancelling the common factors.

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Step 11.2.1

Multiply $9$ by $4$ .

$- \frac{64 π^{2} \sin (\frac{\frac{36 π}{8}}{3})}{9}$

Step 11.2.2

Reduce the expression $\frac{36 π}{8}$ by cancelling the common factors.

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Step 11.2.2.1

Factor $4$ out of $36 π$ .

$- \frac{64 π^{2} \sin (\frac{\frac{4 (9 π)}{8}}{3})}{9}$

Step 11.2.2.2

Factor $4$ out of $8$ .

$- \frac{64 π^{2} \sin (\frac{\frac{4 (9 π)}{4 (2)}}{3})}{9}$

Step 11.2.2.3

Cancel the common factor.

$- \frac{64 π^{2} \sin (\frac{\frac{4 (9 π)}{4 \cdot 2}}{3})}{9}$

Step 11.2.2.4

Rewrite the expression.

$- \frac{64 π^{2} \sin (\frac{\frac{9 π}{2}}{3})}{9}$

Step 11.3

Simplify the numerator.

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Step 11.3.1

Multiply the numerator by the reciprocal of the denominator.

$- \frac{64 π^{2} \sin (\frac{9 π}{2} \cdot \frac{1}{3})}{9}$

Step 11.3.2

Cancel the common factor of $3$ .

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Step 11.3.2.1

Factor $3$ out of $9 π$ .

$- \frac{64 π^{2} \sin (\frac{3 (3 π)}{2} \cdot \frac{1}{3})}{9}$

Step 11.3.2.2

Cancel the common factor.

$- \frac{64 π^{2} \sin (\frac{3 (3 π)}{2} \cdot \frac{1}{3})}{9}$

Step 11.3.2.3

Rewrite the expression.

$- \frac{64 π^{2} \sin (\frac{3 π}{2})}{9}$

Step 11.3.3

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.

$- \frac{64 π^{2} (- \sin (\frac{π}{2}))}{9}$

Step 11.3.4

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

$- \frac{64 π^{2} (- 1 \cdot 1)}{9}$

Step 11.3.5

Multiply $- 1$ by $1$ .

$- \frac{64 π^{2} \cdot - 1}{9}$

Step 11.3.6

Multiply $- 1$ by $64$ .

$- \frac{- 64 π^{2}}{9}$

Step 11.4

Move the negative in front of the fraction.

$- - \frac{64 π^{2}}{9}$

Step 11.5

Multiply $- - \frac{64 π^{2}}{9}$ .

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Step 11.5.1

Multiply $- 1$ by $- 1$ .

$1 \frac{64 π^{2}}{9}$

Step 11.5.2

Multiply $\frac{64 π^{2}}{9}$ by $1$ .

$\frac{64 π^{2}}{9}$

Step 12

$x = \frac{9}{8}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = \frac{9}{8}$ is a local minimum

Step 13

Find the y-value when $x = \frac{9}{8}$ .

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Step 13.1

Replace the variable $x$ with $\frac{9}{8}$ in the expression.

$f (\frac{9}{8}) = 4 \sin (\frac{4 π}{3} \cdot (\frac{9}{8}))$

Step 13.2

Simplify the result.

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Step 13.2.1

Cancel the common factor of $4$ .

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Step 13.2.1.1

Factor $4$ out of $4 π$ .

$f (\frac{9}{8}) = 4 \sin (\frac{4 (π)}{3} \cdot \frac{9}{8})$

Step 13.2.1.2

Factor $4$ out of $8$ .

$f (\frac{9}{8}) = 4 \sin (\frac{4 (π)}{3} \cdot \frac{9}{4 (2)})$

Step 13.2.1.3

Cancel the common factor.

$f (\frac{9}{8}) = 4 \sin (\frac{4 π}{3} \cdot \frac{9}{4 \cdot 2})$

Step 13.2.1.4

Rewrite the expression.

$f (\frac{9}{8}) = 4 \sin (\frac{π}{3} \cdot \frac{9}{2})$

Step 13.2.2

Cancel the common factor of $3$ .

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Step 13.2.2.1

Factor $3$ out of $9$ .

$f (\frac{9}{8}) = 4 \sin (\frac{π}{3} \cdot \frac{3 (3)}{2})$

Step 13.2.2.2

Cancel the common factor.

$f (\frac{9}{8}) = 4 \sin (\frac{π}{3} \cdot \frac{3 \cdot 3}{2})$

Step 13.2.2.3

Rewrite the expression.

$f (\frac{9}{8}) = 4 \sin (π \cdot \frac{3}{2})$

Step 13.2.3

Combine $π$ and $\frac{3}{2}$ .

$f (\frac{9}{8}) = 4 \sin (\frac{π \cdot 3}{2})$

Step 13.2.4

Move $3$ to the left of $π$ .

$f (\frac{9}{8}) = 4 \sin (\frac{3 \cdot π}{2})$

Step 13.2.5

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.

$f (\frac{9}{8}) = 4 (- \sin (\frac{π}{2}))$

Step 13.2.6

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

$f (\frac{9}{8}) = 4 (- 1 \cdot 1)$

Step 13.2.7

Multiply $4 (- 1 \cdot 1)$ .

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Step 13.2.7.1

Multiply $- 1$ by $1$ .

$f (\frac{9}{8}) = 4 \cdot - 1$

Step 13.2.7.2

Multiply $4$ by $- 1$ .

$f (\frac{9}{8}) = - 4$

Step 13.2.8

The final answer is $- 4$ .

$y = - 4$

Step 14

These are the local extrema for $f (x) = 4 \sin (\frac{4 π}{3} x)$ .

$(\frac{3}{8}, 4)$ is a local maxima

$(\frac{9}{8}, - 4)$ is a local minima

Step 15