Calculus Examples

$f (x) = \sin (x) \cos (x)$ , $[0, 2 π]$

Step 1

Find the critical points.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Find the first derivative.

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Step 1.1.1.1

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 $f (x) = \sin (x)$ and $g (x) = \cos (x)$ .

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

Step 1.1.1.2

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

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

Step 1.1.1.3

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

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

Step 1.1.1.4

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

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

Step 1.1.1.5

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

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

Step 1.1.1.6

Add $1$ and $1$ .

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

Step 1.1.1.7

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

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

Step 1.1.1.8

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

$- \sin^{2} (x) + \cos^{1} (x) \cos (x)$

Step 1.1.1.9

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

$- \sin^{2} (x) + \cos^{1} (x) \cos^{1} (x)$

Step 1.1.1.10

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

$- \sin^{2} (x) + {\cos (x)}^{1 + 1}$

Step 1.1.1.11

Add $1$ and $1$ .

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

Step 1.1.1.12

Simplify.

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Step 1.1.1.12.1

Reorder $- \sin^{2} (x)$ and $\cos^{2} (x)$ .

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

Step 1.1.1.12.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \cos (x)$ and $b = \sin (x)$ .

$(\cos (x) + \sin (x)) (\cos (x) - \sin (x))$

Step 1.1.1.12.3

Expand $(\cos (x) + \sin (x)) (\cos (x) - \sin (x))$ using the FOIL Method.

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Step 1.1.1.12.3.1

Apply the distributive property.

$\cos (x) (\cos (x) - \sin (x)) + \sin (x) (\cos (x) - \sin (x))$

Step 1.1.1.12.3.2

Apply the distributive property.

$\cos (x) \cos (x) + \cos (x) (- \sin (x)) + \sin (x) (\cos (x) - \sin (x))$

Step 1.1.1.12.3.3

Apply the distributive property.

$\cos (x) \cos (x) + \cos (x) (- \sin (x)) + \sin (x) \cos (x) + \sin (x) (- \sin (x))$

Step 1.1.1.12.4

Combine the opposite terms in $\cos (x) \cos (x) + \cos (x) (- \sin (x)) + \sin (x) \cos (x) + \sin (x) (- \sin (x))$ .

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Step 1.1.1.12.4.1

Reorder the factors in the terms $\cos (x) (- \sin (x))$ and $\sin (x) \cos (x)$ .

$\cos (x) \cos (x) - \cos (x) \sin (x) + \cos (x) \sin (x) + \sin (x) (- \sin (x))$

Step 1.1.1.12.4.2

Add $- \cos (x) \sin (x)$ and $\cos (x) \sin (x)$ .

$\cos (x) \cos (x) + 0 + \sin (x) (- \sin (x))$

Step 1.1.1.12.4.3

Add $\cos (x) \cos (x)$ and $0$ .

$\cos (x) \cos (x) + \sin (x) (- \sin (x))$

Step 1.1.1.12.5

Simplify each term.

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Step 1.1.1.12.5.1

Multiply $\cos (x) \cos (x)$ .

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Step 1.1.1.12.5.1.1

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

$\cos^{1} (x) \cos (x) + \sin (x) (- \sin (x))$

Step 1.1.1.12.5.1.2

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

$\cos^{1} (x) \cos^{1} (x) + \sin (x) (- \sin (x))$

Step 1.1.1.12.5.1.3

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

${\cos (x)}^{1 + 1} + \sin (x) (- \sin (x))$

Step 1.1.1.12.5.1.4

Add $1$ and $1$ .

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

Step 1.1.1.12.5.2

Rewrite using the commutative property of multiplication.

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

Step 1.1.1.12.5.3

Multiply $- \sin (x) \sin (x)$ .

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Step 1.1.1.12.5.3.1

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

$\cos^{2} (x) - (\sin^{1} (x) \sin (x))$

Step 1.1.1.12.5.3.2

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

$\cos^{2} (x) - (\sin^{1} (x) \sin^{1} (x))$

Step 1.1.1.12.5.3.3

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

$\cos^{2} (x) - {\sin (x)}^{1 + 1}$

Step 1.1.1.12.5.3.4

Add $1$ and $1$ .

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

Step 1.1.1.12.6

Apply the cosine double-angle identity.

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

Step 1.1.2

The first derivative of $f (x)$ with respect to $x$ is $\cos (2 x)$ .

$\cos (2 x)$

Step 1.2

Set the first derivative equal to $0$ then solve the equation $\cos (2 x) = 0$ .

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Step 1.2.1

Set the first derivative equal to $0$ .

$\cos (2 x) = 0$

Step 1.2.2

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

$2 x = \arccos (0)$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

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

$2 x = \frac{π}{2}$

Step 1.2.4

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

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Step 1.2.4.1

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

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

Step 1.2.4.2

Simplify the left side.

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Step 1.2.4.2.1

Cancel the common factor of $2$ .

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Step 1.2.4.2.1.1

Cancel the common factor.

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

Step 1.2.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{π}{2}}{2}$

Step 1.2.4.3

Simplify the right side.

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Step 1.2.4.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{2} \cdot \frac{1}{2}$

Step 1.2.4.3.2

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

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Step 1.2.4.3.2.1

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

$x = \frac{π}{2 \cdot 2}$

Step 1.2.4.3.2.2

Multiply $2$ by $2$ .

$x = \frac{π}{4}$

Step 1.2.5

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.

$2 x = 2 π - \frac{π}{2}$

Step 1.2.6

Solve for $x$ .

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Step 1.2.6.1

Simplify.

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Step 1.2.6.1.1

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

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

Step 1.2.6.1.2

Combine $2 π$ and $\frac{2}{2}$ .

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

Step 1.2.6.1.3

Combine the numerators over the common denominator.

$2 x = \frac{2 π \cdot 2 - π}{2}$

Step 1.2.6.1.4

Multiply $2$ by $2$ .

$2 x = \frac{4 π - π}{2}$

Step 1.2.6.1.5

Subtract $π$ from $4 π$ .

$2 x = \frac{3 π}{2}$

Step 1.2.6.2

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

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Step 1.2.6.2.1

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

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

Step 1.2.6.2.2

Simplify the left side.

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Step 1.2.6.2.2.1

Cancel the common factor of $2$ .

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Step 1.2.6.2.2.1.1

Cancel the common factor.

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

Step 1.2.6.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{3 π}{2}}{2}$

Step 1.2.6.2.3

Simplify the right side.

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Step 1.2.6.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{3 π}{2} \cdot \frac{1}{2}$

Step 1.2.6.2.3.2

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

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Step 1.2.6.2.3.2.1

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

$x = \frac{3 π}{2 \cdot 2}$

Step 1.2.6.2.3.2.2

Multiply $2$ by $2$ .

$x = \frac{3 π}{4}$

Step 1.2.7

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

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Step 1.2.7.1

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

$\frac{2 π}{| b |}$

Step 1.2.7.2

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

$\frac{2 π}{| 2 |}$

Step 1.2.7.3

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

$\frac{2 π}{2}$

Step 1.2.7.4

Cancel the common factor of $2$ .

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Step 1.2.7.4.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 1.2.7.4.2

Divide $π$ by $1$ .

$π$

Step 1.2.8

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

$x = \frac{π}{4} + π n, \frac{3 π}{4} + π n$ , for any integer $n$

Step 1.2.9

Consolidate the answers.

$x = \frac{π}{4} + \frac{π n}{2}$ , for any integer $n$

Step 1.3

Find the values where the derivative is undefined.

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Step 1.3.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 1.4

Evaluate $\sin (x) \cos (x)$ at each $x$ value where the derivative is $0$ or undefined.

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Step 1.4.1

Evaluate at $x = \frac{π}{4}$ .

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Step 1.4.1.1

Substitute $\frac{π}{4}$ for $x$ .

$\sin (\frac{π}{4}) \cos (\frac{π}{4})$

Step 1.4.1.2

Simplify.

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Step 1.4.1.2.1

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$\frac{\sqrt{2}}{2} \cos (\frac{π}{4})$

Step 1.4.1.2.2

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2}}{2}$

Step 1.4.1.2.3

Multiply $\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2}}{2}$ .

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Step 1.4.1.2.3.1

Multiply $\frac{\sqrt{2}}{2}$ by $\frac{\sqrt{2}}{2}$ .

$\frac{\sqrt{2} \sqrt{2}}{2 \cdot 2}$

Step 1.4.1.2.3.2

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{{\sqrt{2}}^{1} \sqrt{2}}{2 \cdot 2}$

Step 1.4.1.2.3.3

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}{2 \cdot 2}$

Step 1.4.1.2.3.4

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

$\frac{{\sqrt{2}}^{1 + 1}}{2 \cdot 2}$

Step 1.4.1.2.3.5

Add $1$ and $1$ .

$\frac{{\sqrt{2}}^{2}}{2 \cdot 2}$

Step 1.4.1.2.3.6

Multiply $2$ by $2$ .

$\frac{{\sqrt{2}}^{2}}{4}$

Step 1.4.1.2.4

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

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Step 1.4.1.2.4.1

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

$\frac{{(2^{\frac{1}{2}})}^{2}}{4}$

Step 1.4.1.2.4.2

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

$\frac{2^{\frac{1}{2} \cdot 2}}{4}$

Step 1.4.1.2.4.3

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

$\frac{2^{\frac{2}{2}}}{4}$

Step 1.4.1.2.4.4

Cancel the common factor of $2$ .

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Step 1.4.1.2.4.4.1

Cancel the common factor.

$\frac{2^{\frac{2}{2}}}{4}$

Step 1.4.1.2.4.4.2

Rewrite the expression.

$\frac{2^{1}}{4}$

Step 1.4.1.2.4.5

Evaluate the exponent.

$\frac{2}{4}$

Step 1.4.1.2.5

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

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Step 1.4.1.2.5.1

Factor $2$ out of $2$ .

$\frac{2 (1)}{4}$

Step 1.4.1.2.5.2

Cancel the common factors.

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Step 1.4.1.2.5.2.1

Factor $2$ out of $4$ .

$\frac{2 \cdot 1}{2 \cdot 2}$

Step 1.4.1.2.5.2.2

Cancel the common factor.

$\frac{2 \cdot 1}{2 \cdot 2}$

Step 1.4.1.2.5.2.3

Rewrite the expression.

$\frac{1}{2}$

Step 1.4.2

Evaluate at $x = \frac{3 π}{4}$ .

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Step 1.4.2.1

Substitute $\frac{3 π}{4}$ for $x$ .

$\sin (\frac{3 π}{4}) \cos (\frac{3 π}{4})$

Step 1.4.2.2

Simplify.

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Step 1.4.2.2.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$\sin (\frac{π}{4}) \cos (\frac{3 π}{4})$

Step 1.4.2.2.2

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$\frac{\sqrt{2}}{2} \cos (\frac{3 π}{4})$

Step 1.4.2.2.3

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$\frac{\sqrt{2}}{2} (- \cos (\frac{π}{4}))$

Step 1.4.2.2.4

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$\frac{\sqrt{2}}{2} (- \frac{\sqrt{2}}{2})$

Step 1.4.2.2.5

Multiply $\frac{\sqrt{2}}{2} (- \frac{\sqrt{2}}{2})$ .

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Step 1.4.2.2.5.1

Multiply $\frac{\sqrt{2}}{2}$ by $\frac{\sqrt{2}}{2}$ .

$- \frac{\sqrt{2} \sqrt{2}}{2 \cdot 2}$

Step 1.4.2.2.5.2

Raise $\sqrt{2}$ to the power of $1$ .

$- \frac{{\sqrt{2}}^{1} \sqrt{2}}{2 \cdot 2}$

Step 1.4.2.2.5.3

Raise $\sqrt{2}$ to the power of $1$ .

$- \frac{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}{2 \cdot 2}$

Step 1.4.2.2.5.4

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

$- \frac{{\sqrt{2}}^{1 + 1}}{2 \cdot 2}$

Step 1.4.2.2.5.5

Add $1$ and $1$ .

$- \frac{{\sqrt{2}}^{2}}{2 \cdot 2}$

Step 1.4.2.2.5.6

Multiply $2$ by $2$ .

$- \frac{{\sqrt{2}}^{2}}{4}$

Step 1.4.2.2.6

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

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Step 1.4.2.2.6.1

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

$- \frac{{(2^{\frac{1}{2}})}^{2}}{4}$

Step 1.4.2.2.6.2

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

$- \frac{2^{\frac{1}{2} \cdot 2}}{4}$

Step 1.4.2.2.6.3

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

$- \frac{2^{\frac{2}{2}}}{4}$

Step 1.4.2.2.6.4

Cancel the common factor of $2$ .

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Step 1.4.2.2.6.4.1

Cancel the common factor.

$- \frac{2^{\frac{2}{2}}}{4}$

Step 1.4.2.2.6.4.2

Rewrite the expression.

$- \frac{2^{1}}{4}$

Step 1.4.2.2.6.5

Evaluate the exponent.

$- \frac{2}{4}$

Step 1.4.2.2.7

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

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Step 1.4.2.2.7.1

Factor $2$ out of $2$ .

$- \frac{2 (1)}{4}$

Step 1.4.2.2.7.2

Cancel the common factors.

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Step 1.4.2.2.7.2.1

Factor $2$ out of $4$ .

$- \frac{2 \cdot 1}{2 \cdot 2}$

Step 1.4.2.2.7.2.2

Cancel the common factor.

$- \frac{2 \cdot 1}{2 \cdot 2}$

Step 1.4.2.2.7.2.3

Rewrite the expression.

$- \frac{1}{2}$

Step 1.4.3

List all of the points.

$(\frac{π}{4} + π n, \frac{1}{2}), (\frac{3 π}{4} + π n, - \frac{1}{2})$ , for any integer $n$

Step 2

Exclude the points that are not on the interval.

$(\frac{π}{4}, \frac{1}{2}), (\frac{5 π}{4}, \frac{1}{2}), (\frac{3 π}{4}, - \frac{1}{2}), (\frac{7 π}{4}, - \frac{1}{2})$

Step 3

Evaluate at the included endpoints.

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Step 3.1

Evaluate at $x = 0$ .

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Step 3.1.1

Substitute $0$ for $x$ .

$\sin (0) \cos (0)$

Step 3.1.2

Simplify.

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Step 3.1.2.1

The exact value of $\sin (0)$ is $0$ .

$0 \cos (0)$

Step 3.1.2.2

The exact value of $\cos (0)$ is $1$ .

$0 \cdot 1$

Step 3.1.2.3

Multiply $0$ by $1$ .

$0$

Step 3.2

Evaluate at $x = 2 π$ .

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Step 3.2.1

Substitute $2 π$ for $x$ .

$\sin (2 π) \cos (2 π)$

Step 3.2.2

Simplify.

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Step 3.2.2.1

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\sin (0) \cos (2 π)$

Step 3.2.2.2

The exact value of $\sin (0)$ is $0$ .

$0 \cos (2 π)$

Step 3.2.2.3

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$0 \cos (0)$

Step 3.2.2.4

The exact value of $\cos (0)$ is $1$ .

$0 \cdot 1$

Step 3.2.2.5

Multiply $0$ by $1$ .

$0$

Step 3.3

List all of the points.

$(0, 0), (2 π, 0)$

Step 4

Compare the $f (x)$ values found for each value of $x$ in order to determine the absolute maximum and minimum over the given interval. The maximum will occur at the highest $f (x)$ value and the minimum will occur at the lowest $f (x)$ value.

Absolute Maximum: $(\frac{π}{4}, \frac{1}{2}), (\frac{5 π}{4}, \frac{1}{2})$

Absolute Minimum: $(\frac{3 π}{4}, - \frac{1}{2}), (\frac{7 π}{4}, - \frac{1}{2})$

Step 5