Calculus Examples

$f (x) = 3 \sin (5 x - 4)$

Step 1

Find the first derivative of the function.

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Step 1.1

Since $3$ is constant with respect to $x$ , the derivative of $3 \sin (5 x - 4)$ with respect to $x$ is $3 \frac{d}{d x} [\sin (5 x - 4)]$ .

$3 \frac{d}{d x} [\sin (5 x - 4)]$

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 $g (x) = 5 x - 4$ .

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Step 1.2.1

To apply the Chain Rule, set $u$ as $5 x - 4$ .

$3 (\frac{d}{d u} [\sin (u)] \frac{d}{d x} [5 x - 4])$

Step 1.2.2

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

$3 (\cos (u) \frac{d}{d x} [5 x - 4])$

Step 1.2.3

Replace all occurrences of $u$ with $5 x - 4$ .

$3 (\cos (5 x - 4) \frac{d}{d x} [5 x - 4])$

Step 1.3

Differentiate.

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Step 1.3.1

By the Sum Rule, the derivative of $5 x - 4$ with respect to $x$ is $\frac{d}{d x} [5 x] + \frac{d}{d x} [- 4]$ .

$3 \cos (5 x - 4) (\frac{d}{d x} [5 x] + \frac{d}{d x} [- 4])$

Step 1.3.2

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$3 \cos (5 x - 4) (5 \frac{d}{d x} [x] + \frac{d}{d x} [- 4])$

Step 1.3.3

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

$3 \cos (5 x - 4) (5 \cdot 1 + \frac{d}{d x} [- 4])$

Step 1.3.4

Multiply $5$ by $1$ .

$3 \cos (5 x - 4) (5 + \frac{d}{d x} [- 4])$

Step 1.3.5

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

$3 \cos (5 x - 4) (5 + 0)$

Step 1.3.6

Simplify the expression.

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Step 1.3.6.1

Add $5$ and $0$ .

$3 \cos (5 x - 4) \cdot 5$

Step 1.3.6.2

Multiply $5$ by $3$ .

$15 \cos (5 x - 4)$

Step 2

Find the second derivative of the function.

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Step 2.1

Since $15$ is constant with respect to $x$ , the derivative of $15 \cos (5 x - 4)$ with respect to $x$ is $15 \frac{d}{d x} [\cos (5 x - 4)]$ .

$f'' (x) = 15 \frac{d}{d x} (\cos (5 x - 4))$

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 $g (x) = 5 x - 4$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $5 x - 4$ .

$f'' (x) = 15 (\frac{d}{d u} (\cos (u)) \frac{d}{d x} (5 x - 4))$

Step 2.2.2

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

$f'' (x) = 15 (- \sin (u) \frac{d}{d x} (5 x - 4))$

Step 2.2.3

Replace all occurrences of $u$ with $5 x - 4$ .

$f'' (x) = 15 (- \sin (5 x - 4) \frac{d}{d x} (5 x - 4))$

Step 2.3

Differentiate.

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Step 2.3.1

Multiply $- 1$ by $15$ .

$f'' (x) = - 15 (\sin (5 x - 4) \frac{d}{d x} (5 x - 4))$

Step 2.3.2

By the Sum Rule, the derivative of $5 x - 4$ with respect to $x$ is $\frac{d}{d x} [5 x] + \frac{d}{d x} [- 4]$ .

$f'' (x) = - 15 \sin (5 x - 4) (\frac{d}{d x} (5 x) + \frac{d}{d x} (- 4))$

Step 2.3.3

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$f'' (x) = - 15 \sin (5 x - 4) (5 \frac{d}{d x} (x) + \frac{d}{d x} (- 4))$

Step 2.3.4

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

$f'' (x) = - 15 \sin (5 x - 4) (5 \cdot 1 + \frac{d}{d x} (- 4))$

Step 2.3.5

Multiply $5$ by $1$ .

$f'' (x) = - 15 \sin (5 x - 4) (5 + \frac{d}{d x} (- 4))$

Step 2.3.6

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

$f'' (x) = - 15 \sin (5 x - 4) (5 + 0)$

Step 2.3.7

Simplify the expression.

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Step 2.3.7.1

Add $5$ and $0$ .

$f'' (x) = - 15 \sin (5 x - 4) \cdot 5$

Step 2.3.7.2

Multiply $5$ by $- 15$ .

$f'' (x) = - 75 \sin (5 x - 4)$

Step 3

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

$15 \cos (5 x - 4) = 0$

Step 4

Divide each term in $15 \cos (5 x - 4) = 0$ by $15$ and simplify.

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Step 4.1

Divide each term in $15 \cos (5 x - 4) = 0$ by $15$ .

$\frac{15 \cos (5 x - 4)}{15} = \frac{0}{15}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $15$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{15 \cos (5 x - 4)}{15} = \frac{0}{15}$

Step 4.2.1.2

Divide $\cos (5 x - 4)$ by $1$ .

$\cos (5 x - 4) = \frac{0}{15}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $0$ by $15$ .

$\cos (5 x - 4) = 0$

Step 5

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

$5 x - 4 = \arccos (0)$

Step 6

Simplify the right side.

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Step 6.1

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

$5 x - 4 = \frac{π}{2}$

Step 7

Add $4$ to both sides of the equation.

$5 x = \frac{π}{2} + 4$

Step 8

Divide each term in $5 x = \frac{π}{2} + 4$ by $5$ and simplify.

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Step 8.1

Divide each term in $5 x = \frac{π}{2} + 4$ by $5$ .

$\frac{5 x}{5} = \frac{\frac{π}{2}}{5} + \frac{4}{5}$

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $5$ .

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Step 8.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{\frac{π}{2}}{5} + \frac{4}{5}$

Step 8.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{π}{2}}{5} + \frac{4}{5}$

Step 8.3

Simplify the right side.

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Step 8.3.1

Simplify each term.

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Step 8.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{2} \cdot \frac{1}{5} + \frac{4}{5}$

Step 8.3.1.2

Multiply $\frac{π}{2} \cdot \frac{1}{5}$ .

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Step 8.3.1.2.1

Multiply $\frac{π}{2}$ by $\frac{1}{5}$ .

$x = \frac{π}{2 \cdot 5} + \frac{4}{5}$

Step 8.3.1.2.2

Multiply $2$ by $5$ .

$x = \frac{π}{10} + \frac{4}{5}$

Step 9

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.

$5 x - 4 = 2 π - \frac{π}{2}$

Step 10

Solve for $x$ .

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Step 10.1

Simplify $2 π - \frac{π}{2}$ .

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Step 10.1.1

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

$5 x - 4 = 2 π \cdot \frac{2}{2} - \frac{π}{2}$

Step 10.1.2

Combine fractions.

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Step 10.1.2.1

Combine $2 π$ and $\frac{2}{2}$ .

$5 x - 4 = \frac{2 π \cdot 2}{2} - \frac{π}{2}$

Step 10.1.2.2

Combine the numerators over the common denominator.

$5 x - 4 = \frac{2 π \cdot 2 - π}{2}$

Step 10.1.3

Simplify the numerator.

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Step 10.1.3.1

Multiply $2$ by $2$ .

$5 x - 4 = \frac{4 π - π}{2}$

Step 10.1.3.2

Subtract $π$ from $4 π$ .

$5 x - 4 = \frac{3 π}{2}$

Step 10.2

Add $4$ to both sides of the equation.

$5 x = \frac{3 π}{2} + 4$

Step 10.3

Divide each term in $5 x = \frac{3 π}{2} + 4$ by $5$ and simplify.

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Step 10.3.1

Divide each term in $5 x = \frac{3 π}{2} + 4$ by $5$ .

$\frac{5 x}{5} = \frac{\frac{3 π}{2}}{5} + \frac{4}{5}$

Step 10.3.2

Simplify the left side.

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Step 10.3.2.1

Cancel the common factor of $5$ .

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Step 10.3.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{\frac{3 π}{2}}{5} + \frac{4}{5}$

Step 10.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{3 π}{2}}{5} + \frac{4}{5}$

Step 10.3.3

Simplify the right side.

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Step 10.3.3.1

Simplify each term.

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Step 10.3.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{3 π}{2} \cdot \frac{1}{5} + \frac{4}{5}$

Step 10.3.3.1.2

Multiply $\frac{3 π}{2} \cdot \frac{1}{5}$ .

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Step 10.3.3.1.2.1

Multiply $\frac{3 π}{2}$ by $\frac{1}{5}$ .

$x = \frac{3 π}{2 \cdot 5} + \frac{4}{5}$

Step 10.3.3.1.2.2

Multiply $2$ by $5$ .

$x = \frac{3 π}{10} + \frac{4}{5}$

Step 11

The solution to the equation $5 x - 4 = \frac{π}{2}$ .

$x = \frac{π}{10} + \frac{4}{5}, \frac{3 π}{10} + \frac{4}{5}$

Step 12

Evaluate the second derivative at $x = \frac{π}{10} + \frac{4}{5}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- 75 \sin (5 (\frac{π}{10} + \frac{4}{5}) - 4)$

Step 13

Evaluate the second derivative.

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Step 13.1

Simplify each term.

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Step 13.1.1

Apply the distributive property.

$- 75 \sin (5 \frac{π}{10} + 5 (\frac{4}{5}) - 4)$

Step 13.1.2

Cancel the common factor of $5$ .

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Step 13.1.2.1

Factor $5$ out of $10$ .

$- 75 \sin (5 \frac{π}{5 (2)} + 5 (\frac{4}{5}) - 4)$

Step 13.1.2.2

Cancel the common factor.

$- 75 \sin (5 \frac{π}{5 \cdot 2} + 5 (\frac{4}{5}) - 4)$

Step 13.1.2.3

Rewrite the expression.

$- 75 \sin (\frac{π}{2} + 5 (\frac{4}{5}) - 4)$

Step 13.1.3

Cancel the common factor of $5$ .

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Step 13.1.3.1

Cancel the common factor.

$- 75 \sin (\frac{π}{2} + 5 (\frac{4}{5}) - 4)$

Step 13.1.3.2

Rewrite the expression.

$- 75 \sin (\frac{π}{2} + 4 - 4)$

Step 13.2

Simplify by subtracting numbers.

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Step 13.2.1

Subtract $4$ from $4$ .

$- 75 \sin (\frac{π}{2} + 0)$

Step 13.2.2

Add $\frac{π}{2}$ and $0$ .

$- 75 \sin (\frac{π}{2})$

Step 13.3

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

$- 75 \cdot 1$

Step 13.4

Multiply $- 75$ by $1$ .

$- 75$

Step 14

$x = \frac{π}{10} + \frac{4}{5}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = \frac{π}{10} + \frac{4}{5}$ is a local maximum

Step 15

Find the y-value when $x = \frac{π}{10} + \frac{4}{5}$ .

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Step 15.1

Replace the variable $x$ with $\frac{π}{10} + \frac{4}{5}$ in the expression.

$f (\frac{π}{10} + \frac{4}{5}) = 3 \sin (5 (\frac{π}{10} + \frac{4}{5}) - 4)$

Step 15.2

Simplify the result.

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Step 15.2.1

Simplify each term.

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Step 15.2.1.1

Apply the distributive property.

$f (\frac{π}{10} + \frac{4}{5}) = 3 \sin (5 (\frac{π}{10}) + 5 (\frac{4}{5}) - 4)$

Step 15.2.1.2

Cancel the common factor of $5$ .

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Step 15.2.1.2.1

Factor $5$ out of $10$ .

$f (\frac{π}{10} + \frac{4}{5}) = 3 \sin (5 (\frac{π}{5 (2)}) + 5 (\frac{4}{5}) - 4)$

Step 15.2.1.2.2

Cancel the common factor.

$f (\frac{π}{10} + \frac{4}{5}) = 3 \sin (5 (\frac{π}{5 \cdot 2}) + 5 (\frac{4}{5}) - 4)$

Step 15.2.1.2.3

Rewrite the expression.

$f (\frac{π}{10} + \frac{4}{5}) = 3 \sin (\frac{π}{2} + 5 (\frac{4}{5}) - 4)$

Step 15.2.1.3

Cancel the common factor of $5$ .

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Step 15.2.1.3.1

Cancel the common factor.

$f (\frac{π}{10} + \frac{4}{5}) = 3 \sin (\frac{π}{2} + 5 (\frac{4}{5}) - 4)$

Step 15.2.1.3.2

Rewrite the expression.

$f (\frac{π}{10} + \frac{4}{5}) = 3 \sin (\frac{π}{2} + 4 - 4)$

Step 15.2.2

Simplify by subtracting numbers.

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Step 15.2.2.1

Subtract $4$ from $4$ .

$f (\frac{π}{10} + \frac{4}{5}) = 3 \sin (\frac{π}{2} + 0)$

Step 15.2.2.2

Add $\frac{π}{2}$ and $0$ .

$f (\frac{π}{10} + \frac{4}{5}) = 3 \sin (\frac{π}{2})$

Step 15.2.3

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

$f (\frac{π}{10} + \frac{4}{5}) = 3 \cdot 1$

Step 15.2.4

Multiply $3$ by $1$ .

$f (\frac{π}{10} + \frac{4}{5}) = 3$

Step 15.2.5

The final answer is $3$ .

$y = 3$

Step 16

Evaluate the second derivative at $x = \frac{3 π}{10} + \frac{4}{5}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- 75 \sin (5 (\frac{3 π}{10} + \frac{4}{5}) - 4)$

Step 17

Evaluate the second derivative.

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Step 17.1

Simplify each term.

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Step 17.1.1

Apply the distributive property.

$- 75 \sin (5 \frac{3 π}{10} + 5 (\frac{4}{5}) - 4)$

Step 17.1.2

Cancel the common factor of $5$ .

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Step 17.1.2.1

Factor $5$ out of $10$ .

$- 75 \sin (5 \frac{3 π}{5 (2)} + 5 (\frac{4}{5}) - 4)$

Step 17.1.2.2

Cancel the common factor.

$- 75 \sin (5 \frac{3 π}{5 \cdot 2} + 5 (\frac{4}{5}) - 4)$

Step 17.1.2.3

Rewrite the expression.

$- 75 \sin (\frac{3 π}{2} + 5 (\frac{4}{5}) - 4)$

Step 17.1.3

Cancel the common factor of $5$ .

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Step 17.1.3.1

Cancel the common factor.

$- 75 \sin (\frac{3 π}{2} + 5 (\frac{4}{5}) - 4)$

Step 17.1.3.2

Rewrite the expression.

$- 75 \sin (\frac{3 π}{2} + 4 - 4)$

Step 17.2

Simplify by subtracting numbers.

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Step 17.2.1

Subtract $4$ from $4$ .

$- 75 \sin (\frac{3 π}{2} + 0)$

Step 17.2.2

Add $\frac{3 π}{2}$ and $0$ .

$- 75 \sin (\frac{3 π}{2})$

Step 17.3

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.

$- 75 (- \sin (\frac{π}{2}))$

Step 17.4

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

$- 75 (- 1 \cdot 1)$

Step 17.5

Multiply $- 75 (- 1 \cdot 1)$ .

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Step 17.5.1

Multiply $- 1$ by $1$ .

$- 75 \cdot - 1$

Step 17.5.2

Multiply $- 75$ by $- 1$ .

$75$

Step 18

$x = \frac{3 π}{10} + \frac{4}{5}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = \frac{3 π}{10} + \frac{4}{5}$ is a local minimum

Step 19

Find the y-value when $x = \frac{3 π}{10} + \frac{4}{5}$ .

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Step 19.1

Replace the variable $x$ with $\frac{3 π}{10} + \frac{4}{5}$ in the expression.

$f (\frac{3 π}{10} + \frac{4}{5}) = 3 \sin (5 (\frac{3 π}{10} + \frac{4}{5}) - 4)$

Step 19.2

Simplify the result.

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Step 19.2.1

Simplify each term.

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Step 19.2.1.1

Apply the distributive property.

$f (\frac{3 π}{10} + \frac{4}{5}) = 3 \sin (5 (\frac{3 π}{10}) + 5 (\frac{4}{5}) - 4)$

Step 19.2.1.2

Cancel the common factor of $5$ .

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Step 19.2.1.2.1

Factor $5$ out of $10$ .

$f (\frac{3 π}{10} + \frac{4}{5}) = 3 \sin (5 (\frac{3 π}{5 (2)}) + 5 (\frac{4}{5}) - 4)$

Step 19.2.1.2.2

Cancel the common factor.

$f (\frac{3 π}{10} + \frac{4}{5}) = 3 \sin (5 (\frac{3 π}{5 \cdot 2}) + 5 (\frac{4}{5}) - 4)$

Step 19.2.1.2.3

Rewrite the expression.

$f (\frac{3 π}{10} + \frac{4}{5}) = 3 \sin (\frac{3 π}{2} + 5 (\frac{4}{5}) - 4)$

Step 19.2.1.3

Cancel the common factor of $5$ .

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Step 19.2.1.3.1

Cancel the common factor.

$f (\frac{3 π}{10} + \frac{4}{5}) = 3 \sin (\frac{3 π}{2} + 5 (\frac{4}{5}) - 4)$

Step 19.2.1.3.2

Rewrite the expression.

$f (\frac{3 π}{10} + \frac{4}{5}) = 3 \sin (\frac{3 π}{2} + 4 - 4)$

Step 19.2.2

Simplify by subtracting numbers.

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Step 19.2.2.1

Subtract $4$ from $4$ .

$f (\frac{3 π}{10} + \frac{4}{5}) = 3 \sin (\frac{3 π}{2} + 0)$

Step 19.2.2.2

Add $\frac{3 π}{2}$ and $0$ .

$f (\frac{3 π}{10} + \frac{4}{5}) = 3 \sin (\frac{3 π}{2})$

Step 19.2.3

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.

$f (\frac{3 π}{10} + \frac{4}{5}) = 3 (- \sin (\frac{π}{2}))$

Step 19.2.4

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

$f (\frac{3 π}{10} + \frac{4}{5}) = 3 (- 1 \cdot 1)$

Step 19.2.5

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

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Step 19.2.5.1

Multiply $- 1$ by $1$ .

$f (\frac{3 π}{10} + \frac{4}{5}) = 3 \cdot - 1$

Step 19.2.5.2

Multiply $3$ by $- 1$ .

$f (\frac{3 π}{10} + \frac{4}{5}) = - 3$

Step 19.2.6

The final answer is $- 3$ .

$y = - 3$

Step 20

These are the local extrema for $f (x) = 3 \sin (5 x - 4)$ .

$(\frac{π}{10} + \frac{4}{5}, 3)$ is a local maxima

$(\frac{3 π}{10} + \frac{4}{5}, - 3)$ is a local minima

Step 21