Calculus Examples

$y = 3 \sin (2 x - \frac{π}{2}) + 1$

Step 1

Find the first derivative of the function.

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Step 1.1

By the Sum Rule, the derivative of $3 \sin (2 x - \frac{π}{2}) + 1$ with respect to $x$ is $\frac{d}{d x} [3 \sin (2 x - \frac{π}{2})] + \frac{d}{d x} [1]$ .

$\frac{d}{d x} [3 \sin (2 x - \frac{π}{2})] + \frac{d}{d x} [1]$

Step 1.2

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

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Step 1.2.1

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

$3 \frac{d}{d x} [\sin (2 x - \frac{π}{2})] + \frac{d}{d x} [1]$

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 $f (x) = \sin (x)$ and $g (x) = 2 x - \frac{π}{2}$ .

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Step 1.2.2.1

To apply the Chain Rule, set $u$ as $2 x - \frac{π}{2}$ .

$3 (\frac{d}{d u} [\sin (u)] \frac{d}{d x} [2 x - \frac{π}{2}]) + \frac{d}{d x} [1]$

Step 1.2.2.2

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

$3 (\cos (u) \frac{d}{d x} [2 x - \frac{π}{2}]) + \frac{d}{d x} [1]$

Step 1.2.2.3

Replace all occurrences of $u$ with $2 x - \frac{π}{2}$ .

$3 (\cos (2 x - \frac{π}{2}) \frac{d}{d x} [2 x - \frac{π}{2}]) + \frac{d}{d x} [1]$

Step 1.2.3

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

$3 (\cos (2 x - \frac{π}{2}) (\frac{d}{d x} [2 x] + \frac{d}{d x} [- \frac{π}{2}])) + \frac{d}{d x} [1]$

Step 1.2.4

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

$3 (\cos (2 x - \frac{π}{2}) (2 \frac{d}{d x} [x] + \frac{d}{d x} [- \frac{π}{2}])) + \frac{d}{d x} [1]$

Step 1.2.5

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

$3 (\cos (2 x - \frac{π}{2}) (2 \cdot 1 + \frac{d}{d x} [- \frac{π}{2}])) + \frac{d}{d x} [1]$

Step 1.2.6

Since $- \frac{π}{2}$ is constant with respect to $x$ , the derivative of $- \frac{π}{2}$ with respect to $x$ is $0$ .

$3 (\cos (2 x - \frac{π}{2}) (2 \cdot 1 + 0)) + \frac{d}{d x} [1]$

Step 1.2.7

Multiply $2$ by $1$ .

$3 (\cos (2 x - \frac{π}{2}) (2 + 0)) + \frac{d}{d x} [1]$

Step 1.2.8

Add $2$ and $0$ .

$3 (\cos (2 x - \frac{π}{2}) \cdot 2) + \frac{d}{d x} [1]$

Step 1.2.9

Move $2$ to the left of $\cos (2 x - \frac{π}{2})$ .

$3 (2 \cdot \cos (2 x - \frac{π}{2})) + \frac{d}{d x} [1]$

Step 1.2.10

Multiply $2$ by $3$ .

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

Step 1.3

Differentiate using the Constant Rule.

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Step 1.3.1

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$6 \cos (2 x - \frac{π}{2}) + 0$

Step 1.3.2

Add $6 \cos (2 x - \frac{π}{2})$ and $0$ .

$6 \cos (2 x - \frac{π}{2})$

Step 2

Find the second derivative of the function.

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Step 2.1

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

$f'' (x) = 6 \frac{d}{d x} (\cos (2 x - \frac{π}{2}))$

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 $g (x) = 2 x - \frac{π}{2}$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $2 x - \frac{π}{2}$ .

$f'' (x) = 6 (\frac{d}{d u} (\cos (u)) \frac{d}{d x} (2 x - \frac{π}{2}))$

Step 2.2.2

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

$f'' (x) = 6 (- \sin (u) \frac{d}{d x} (2 x - \frac{π}{2}))$

Step 2.2.3

Replace all occurrences of $u$ with $2 x - \frac{π}{2}$ .

$f'' (x) = 6 (- \sin (2 x - \frac{π}{2}) \frac{d}{d x} (2 x - \frac{π}{2}))$

Step 2.3

Differentiate.

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Step 2.3.1

Multiply $- 1$ by $6$ .

$f'' (x) = - 6 (\sin (2 x - \frac{π}{2}) \frac{d}{d x} (2 x - \frac{π}{2}))$

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.3.4

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

$f'' (x) = - 6 \sin (2 x - \frac{π}{2}) (2 \cdot 1 + \frac{d}{d x} (- \frac{π}{2}))$

Step 2.3.5

Multiply $2$ by $1$ .

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

Step 2.3.6

Since $- \frac{π}{2}$ is constant with respect to $x$ , the derivative of $- \frac{π}{2}$ with respect to $x$ is $0$ .

$f'' (x) = - 6 \sin (2 x - \frac{π}{2}) (2 + 0)$

Step 2.3.7

Simplify the expression.

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Step 2.3.7.1

Add $2$ and $0$ .

$f'' (x) = - 6 \sin (2 x - \frac{π}{2}) \cdot 2$

Step 2.3.7.2

Multiply $2$ by $- 6$ .

$f'' (x) = - 12 \sin (2 x - \frac{π}{2})$

Step 3

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

$6 \cos (2 x - \frac{π}{2}) = 0$

Step 4

Divide each term in $6 \cos (2 x - \frac{π}{2}) = 0$ by $6$ and simplify.

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Step 4.1

Divide each term in $6 \cos (2 x - \frac{π}{2}) = 0$ by $6$ .

$\frac{6 \cos (2 x - \frac{π}{2})}{6} = \frac{0}{6}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $6$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{6 \cos (2 x - \frac{π}{2})}{6} = \frac{0}{6}$

Step 4.2.1.2

Divide $\cos (2 x - \frac{π}{2})$ by $1$ .

$\cos (2 x - \frac{π}{2}) = \frac{0}{6}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $0$ by $6$ .

$\cos (2 x - \frac{π}{2}) = 0$

Step 5

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

$2 x - \frac{π}{2} = \arccos (0)$

Step 6

Simplify the right side.

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Step 6.1

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

$2 x - \frac{π}{2} = \frac{π}{2}$

Step 7

Move all terms not containing $x$ to the right side of the equation.

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Step 7.1

Add $\frac{π}{2}$ to both sides of the equation.

$2 x = \frac{π}{2} + \frac{π}{2}$

Step 7.2

Combine the numerators over the common denominator.

$2 x = \frac{π + π}{2}$

Step 7.3

Add $π$ and $π$ .

$2 x = \frac{2 π}{2}$

Step 7.4

Cancel the common factor of $2$ .

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Step 7.4.1

Cancel the common factor.

$2 x = \frac{2 π}{2}$

Step 7.4.2

Divide $π$ by $1$ .

$2 x = π$

Step 8

Divide each term in $2 x = π$ by $2$ and simplify.

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Step 8.1

Divide each term in $2 x = π$ by $2$ .

$\frac{2 x}{2} = \frac{π}{2}$

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $2$ .

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Step 8.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{π}{2}$

Step 8.2.1.2

Divide $x$ by $1$ .

$x = \frac{π}{2}$

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.

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

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 $\frac{2}{2}$ .

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

Step 10.1.2

Combine fractions.

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Step 10.1.2.1

Combine $2 π$ and $\frac{2}{2}$ .

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

Step 10.1.2.2

Combine the numerators over the common denominator.

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

Step 10.1.3

Simplify the numerator.

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Step 10.1.3.1

Multiply $2$ by $2$ .

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

Step 10.1.3.2

Subtract $π$ from $4 π$ .

$2 x - \frac{π}{2} = \frac{3 π}{2}$

Step 10.2

Move all terms not containing $x$ to the right side of the equation.

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Step 10.2.1

Add $\frac{π}{2}$ to both sides of the equation.

$2 x = \frac{3 π}{2} + \frac{π}{2}$

Step 10.2.2

Combine the numerators over the common denominator.

$2 x = \frac{3 π + π}{2}$

Step 10.2.3

Add $3 π$ and $π$ .

$2 x = \frac{4 π}{2}$

Step 10.2.4

Cancel the common factor of $4$ and $2$ .

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Step 10.2.4.1

Factor $2$ out of $4 π$ .

$2 x = \frac{2 (2 π)}{2}$

Step 10.2.4.2

Cancel the common factors.

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Step 10.2.4.2.1

Factor $2$ out of $2$ .

$2 x = \frac{2 (2 π)}{2 (1)}$

Step 10.2.4.2.2

Cancel the common factor.

$2 x = \frac{2 (2 π)}{2 \cdot 1}$

Step 10.2.4.2.3

Rewrite the expression.

$2 x = \frac{2 π}{1}$

Step 10.2.4.2.4

Divide $2 π$ by $1$ .

$2 x = 2 π$

Step 10.3

Divide each term in $2 x = 2 π$ by $2$ and simplify.

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Step 10.3.1

Divide each term in $2 x = 2 π$ by $2$ .

$\frac{2 x}{2} = \frac{2 π}{2}$

Step 10.3.2

Simplify the left side.

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Step 10.3.2.1

Cancel the common factor of $2$ .

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Step 10.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{2 π}{2}$

Step 10.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{2 π}{2}$

Step 10.3.3

Simplify the right side.

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Step 10.3.3.1

Cancel the common factor of $2$ .

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Step 10.3.3.1.1

Cancel the common factor.

$x = \frac{2 π}{2}$

Step 10.3.3.1.2

Divide $π$ by $1$ .

$x = π$

Step 11

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

$x = \frac{π}{2}, π$

Step 12

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.

$- 12 \sin (2 (\frac{π}{2}) - \frac{π}{2})$

Step 13

Evaluate the second derivative.

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Step 13.1

Cancel the common factor of $2$ .

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Step 13.1.1

Cancel the common factor.

$- 12 \sin (2 \frac{π}{2} - \frac{π}{2})$

Step 13.1.2

Rewrite the expression.

$- 12 \sin (π - \frac{π}{2})$

Step 13.2

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

$- 12 \sin (π \cdot \frac{2}{2} - \frac{π}{2})$

Step 13.3

Combine fractions.

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Step 13.3.1

Combine $π$ and $\frac{2}{2}$ .

$- 12 \sin (\frac{π \cdot 2}{2} - \frac{π}{2})$

Step 13.3.2

Combine the numerators over the common denominator.

$- 12 \sin (\frac{π \cdot 2 - π}{2})$

Step 13.4

Simplify the numerator.

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Step 13.4.1

Move $2$ to the left of $π$ .

$- 12 \sin (\frac{2 \cdot π - π}{2})$

Step 13.4.2

Subtract $π$ from $2 π$ .

$- 12 \sin (\frac{π}{2})$

Step 13.5

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

$- 12 \cdot 1$

Step 13.6

Multiply $- 12$ by $1$ .

$- 12$

Step 14

$x = \frac{π}{2}$ 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 15

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

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Step 15.1

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

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

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

Cancel the common factor of $2$ .

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Step 15.2.1.1.1

Cancel the common factor.

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

Step 15.2.1.1.2

Rewrite the expression.

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

Step 15.2.1.2

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

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

Step 15.2.1.3

Combine $π$ and $\frac{2}{2}$ .

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

Step 15.2.1.4

Combine the numerators over the common denominator.

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

Step 15.2.1.5

Simplify the numerator.

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Step 15.2.1.5.1

Move $2$ to the left of $π$ .

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

Step 15.2.1.5.2

Subtract $π$ from $2 π$ .

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

Step 15.2.1.6

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

$f (\frac{π}{2}) = 3 \cdot 1 + 1$

Step 15.2.1.7

Multiply $3$ by $1$ .

$f (\frac{π}{2}) = 3 + 1$

Step 15.2.2

Add $3$ and $1$ .

$f (\frac{π}{2}) = 4$

Step 15.2.3

The final answer is $4$ .

$y = 4$

Step 16

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

$- 12 \sin (2 (π) - \frac{π}{2})$

Step 17

Evaluate the second derivative.

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Step 17.1

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

$- 12 \sin (2 π \cdot \frac{2}{2} - \frac{π}{2})$

Step 17.2

Combine fractions.

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Step 17.2.1

Combine $2 π$ and $\frac{2}{2}$ .

$- 12 \sin (\frac{2 π \cdot 2}{2} - \frac{π}{2})$

Step 17.2.2

Combine the numerators over the common denominator.

$- 12 \sin (\frac{2 π \cdot 2 - π}{2})$

Step 17.3

Simplify the numerator.

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Step 17.3.1

Multiply $2$ by $2$ .

$- 12 \sin (\frac{4 π - π}{2})$

Step 17.3.2

Subtract $π$ from $4 π$ .

$- 12 \sin (\frac{3 π}{2})$

Step 17.4

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.

$- 12 (- \sin (\frac{π}{2}))$

Step 17.5

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

$- 12 (- 1 \cdot 1)$

Step 17.6

Multiply $- 12 (- 1 \cdot 1)$ .

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Step 17.6.1

Multiply $- 1$ by $1$ .

$- 12 \cdot - 1$

Step 17.6.2

Multiply $- 12$ by $- 1$ .

$12$

Step 18

$x = π$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = π$ is a local minimum

Step 19

Find the y-value when $x = π$ .

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Step 19.1

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

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

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

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

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

Step 19.2.1.2

Combine $2 π$ and $\frac{2}{2}$ .

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

Step 19.2.1.3

Combine the numerators over the common denominator.

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

Step 19.2.1.4

Simplify the numerator.

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Step 19.2.1.4.1

Multiply $2$ by $2$ .

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

Step 19.2.1.4.2

Subtract $π$ from $4 π$ .

$f (π) = 3 \sin (\frac{3 π}{2}) + 1$

Step 19.2.1.5

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.

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

Step 19.2.1.6

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

$f (π) = 3 (- 1 \cdot 1) + 1$

Step 19.2.1.7

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

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Step 19.2.1.7.1

Multiply $- 1$ by $1$ .

$f (π) = 3 \cdot - 1 + 1$

Step 19.2.1.7.2

Multiply $3$ by $- 1$ .

$f (π) = - 3 + 1$

Step 19.2.2

Add $- 3$ and $1$ .

$f (π) = - 2$

Step 19.2.3

The final answer is $- 2$ .

$y = - 2$

Step 20

These are the local extrema for $f (x) = 3 \sin (2 x - \frac{π}{2}) + 1$ .

$(\frac{π}{2}, 4)$ is a local maxima

$(π, - 2)$ is a local minima

Step 21