Calculus Examples

$f (x) = \sin (x) \cos (x) + 7$

Step 1

Find the first derivative of the function.

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Step 1.1

By the Sum Rule, the derivative of $\sin (x) \cos (x) + 7$ with respect to $x$ is $\frac{d}{d x} [\sin (x) \cos (x)] + \frac{d}{d x} [7]$ .

$\frac{d}{d x} [\sin (x) \cos (x)] + \frac{d}{d x} [7]$

Step 1.2

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

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Step 1.2.1

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 $g (x) = \cos (x)$ .

$\sin (x) \frac{d}{d x} [\cos (x)] + \cos (x) \frac{d}{d x} [\sin (x)] + \frac{d}{d x} [7]$

Step 1.2.2

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

$\sin (x) (- \sin (x)) + \cos (x) \frac{d}{d x} [\sin (x)] + \frac{d}{d x} [7]$

Step 1.2.3

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

$\sin (x) (- \sin (x)) + \cos (x) \cos (x) + \frac{d}{d x} [7]$

Step 1.2.4

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

$- (\sin^{1} (x) \sin (x)) + \cos (x) \cos (x) + \frac{d}{d x} [7]$

Step 1.2.5

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

$- (\sin^{1} (x) \sin^{1} (x)) + \cos (x) \cos (x) + \frac{d}{d x} [7]$

Step 1.2.6

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

$- {\sin (x)}^{1 + 1} + \cos (x) \cos (x) + \frac{d}{d x} [7]$

Step 1.2.7

Add $1$ and $1$ .

$- \sin^{2} (x) + \cos (x) \cos (x) + \frac{d}{d x} [7]$

Step 1.2.8

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

$- \sin^{2} (x) + \cos^{1} (x) \cos (x) + \frac{d}{d x} [7]$

Step 1.2.9

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

$- \sin^{2} (x) + \cos^{1} (x) \cos^{1} (x) + \frac{d}{d x} [7]$

Step 1.2.10

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

$- \sin^{2} (x) + {\cos (x)}^{1 + 1} + \frac{d}{d x} [7]$

Step 1.2.11

Add $1$ and $1$ .

$- \sin^{2} (x) + \cos^{2} (x) + \frac{d}{d x} [7]$

Step 1.3

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$- \sin^{2} (x) + \cos^{2} (x) + 0$

Step 1.4

Simplify.

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Step 1.4.1

Add $- \sin^{2} (x) + \cos^{2} (x)$ and $0$ .

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

Step 1.4.2

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

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

Step 1.4.3

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 1.4.4

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

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Step 1.4.4.1

Apply the distributive property.

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

Step 1.4.4.2

Apply the distributive property.

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

Step 1.4.4.3

Apply the distributive property.

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

Step 1.4.5

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 1.4.5.1

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 1.4.5.2

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

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

Step 1.4.5.3

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

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

Step 1.4.6

Simplify each term.

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Step 1.4.6.1

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

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Step 1.4.6.1.1

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

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

Step 1.4.6.1.2

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

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

Step 1.4.6.1.3

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

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

Step 1.4.6.1.4

Add $1$ and $1$ .

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

Step 1.4.6.2

Rewrite using the commutative property of multiplication.

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

Step 1.4.6.3

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

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Step 1.4.6.3.1

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

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

Step 1.4.6.3.2

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

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

Step 1.4.6.3.3

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

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

Step 1.4.6.3.4

Add $1$ and $1$ .

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

Step 1.4.7

Apply the cosine double-angle identity.

$\cos (2 x)$

Step 2

Find the second derivative of the function.

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Step 2.1

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 2.1.1

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

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

Step 2.1.2

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

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

Step 2.1.3

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

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

Step 2.2

Differentiate.

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Step 2.2.1

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

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

Step 2.2.2

Multiply $2$ by $- 1$ .

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

Step 2.2.3

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

$f'' (x) = - 2 \sin (2 x) \cdot 1$

Step 2.2.4

Multiply $- 2$ by $1$ .

$f'' (x) = - 2 \sin (2 x)$

Step 3

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

$\cos (2 x) = 0$

Step 4

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

$2 x = \arccos (0)$

Step 5

Simplify the right side.

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Step 5.1

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

$2 x = \frac{π}{2}$

Step 6

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

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Step 6.1

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

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

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $2$ .

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Step 6.2.1.1

Cancel the common factor.

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

Step 6.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{π}{2}}{2}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{2} \cdot \frac{1}{2}$

Step 6.3.2

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

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Step 6.3.2.1

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

$x = \frac{π}{2 \cdot 2}$

Step 6.3.2.2

Multiply $2$ by $2$ .

$x = \frac{π}{4}$

Step 7

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 8

Solve for $x$ .

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Step 8.1

Simplify.

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Step 8.1.1

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 8.1.2

Combine $2 π$ and $\frac{2}{2}$ .

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

Step 8.1.3

Combine the numerators over the common denominator.

$2 x = \frac{2 π \cdot 2 - π}{2}$

Step 8.1.4

Multiply $2$ by $2$ .

$2 x = \frac{4 π - π}{2}$

Step 8.1.5

Subtract $π$ from $4 π$ .

$2 x = \frac{3 π}{2}$

Step 8.2

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

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Step 8.2.1

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

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

Step 8.2.2

Simplify the left side.

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Step 8.2.2.1

Cancel the common factor of $2$ .

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Step 8.2.2.1.1

Cancel the common factor.

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

Step 8.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{3 π}{2}}{2}$

Step 8.2.3

Simplify the right side.

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Step 8.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{3 π}{2} \cdot \frac{1}{2}$

Step 8.2.3.2

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

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Step 8.2.3.2.1

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

$x = \frac{3 π}{2 \cdot 2}$

Step 8.2.3.2.2

Multiply $2$ by $2$ .

$x = \frac{3 π}{4}$

Step 9

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

$x = \frac{π}{4}, \frac{3 π}{4}$

Step 10

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

$- 2 \sin (2 (\frac{π}{4}))$

Step 11

Evaluate the second derivative.

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Step 11.1

Cancel the common factor of $2$ .

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Step 11.1.1

Factor $2$ out of $4$ .

$- 2 \sin (2 \frac{π}{2 (2)})$

Step 11.1.2

Cancel the common factor.

$- 2 \sin (2 \frac{π}{2 \cdot 2})$

Step 11.1.3

Rewrite the expression.

$- 2 \sin (\frac{π}{2})$

Step 11.2

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

$- 2 \cdot 1$

Step 11.3

Multiply $- 2$ by $1$ .

$- 2$

Step 12

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

$x = \frac{π}{4}$ is a local maximum

Step 13

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

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Step 13.1

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

$f (\frac{π}{4}) = \sin (\frac{π}{4}) \cos (\frac{π}{4}) + 7$

Step 13.2

Simplify the result.

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Step 13.2.1

Simplify each term.

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Step 13.2.1.1

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

$f (\frac{π}{4}) = \frac{\sqrt{2}}{2} \cdot \cos (\frac{π}{4}) + 7$

Step 13.2.1.2

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

$f (\frac{π}{4}) = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2}}{2} + 7$

Step 13.2.1.3

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

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Step 13.2.1.3.1

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

$f (\frac{π}{4}) = \frac{\sqrt{2} \sqrt{2}}{2 \cdot 2} + 7$

Step 13.2.1.3.2

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

$f (\frac{π}{4}) = \frac{\sqrt{2} \sqrt{2}}{2 \cdot 2} + 7$

Step 13.2.1.3.3

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

$f (\frac{π}{4}) = \frac{\sqrt{2} \sqrt{2}}{2 \cdot 2} + 7$

Step 13.2.1.3.4

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

$f (\frac{π}{4}) = \frac{{\sqrt{2}}^{1 + 1}}{2 \cdot 2} + 7$

Step 13.2.1.3.5

Add $1$ and $1$ .

$f (\frac{π}{4}) = \frac{{\sqrt{2}}^{2}}{2 \cdot 2} + 7$

Step 13.2.1.3.6

Multiply $2$ by $2$ .

$f (\frac{π}{4}) = \frac{{\sqrt{2}}^{2}}{4} + 7$

Step 13.2.1.4

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

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Step 13.2.1.4.1

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

$f (\frac{π}{4}) = \frac{{(2^{\frac{1}{2}})}^{2}}{4} + 7$

Step 13.2.1.4.2

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

$f (\frac{π}{4}) = \frac{2^{\frac{1}{2} \cdot 2}}{4} + 7$

Step 13.2.1.4.3

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

$f (\frac{π}{4}) = \frac{2^{\frac{2}{2}}}{4} + 7$

Step 13.2.1.4.4

Cancel the common factor of $2$ .

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Step 13.2.1.4.4.1

Cancel the common factor.

$f (\frac{π}{4}) = \frac{2^{\frac{2}{2}}}{4} + 7$

Step 13.2.1.4.4.2

Rewrite the expression.

$f (\frac{π}{4}) = \frac{2}{4} + 7$

Step 13.2.1.4.5

Evaluate the exponent.

$f (\frac{π}{4}) = \frac{2}{4} + 7$

Step 13.2.1.5

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

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Step 13.2.1.5.1

Factor $2$ out of $2$ .

$f (\frac{π}{4}) = \frac{2 (1)}{4} + 7$

Step 13.2.1.5.2

Cancel the common factors.

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Step 13.2.1.5.2.1

Factor $2$ out of $4$ .

$f (\frac{π}{4}) = \frac{2 \cdot 1}{2 \cdot 2} + 7$

Step 13.2.1.5.2.2

Cancel the common factor.

$f (\frac{π}{4}) = \frac{2 \cdot 1}{2 \cdot 2} + 7$

Step 13.2.1.5.2.3

Rewrite the expression.

$f (\frac{π}{4}) = \frac{1}{2} + 7$

Step 13.2.2

To write $7$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (\frac{π}{4}) = \frac{1}{2} + 7 \cdot \frac{2}{2}$

Step 13.2.3

Combine $7$ and $\frac{2}{2}$ .

$f (\frac{π}{4}) = \frac{1}{2} + \frac{7 \cdot 2}{2}$

Step 13.2.4

Combine the numerators over the common denominator.

$f (\frac{π}{4}) = \frac{1 + 7 \cdot 2}{2}$

Step 13.2.5

Simplify the numerator.

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Step 13.2.5.1

Multiply $7$ by $2$ .

$f (\frac{π}{4}) = \frac{1 + 14}{2}$

Step 13.2.5.2

Add $1$ and $14$ .

$f (\frac{π}{4}) = \frac{15}{2}$

Step 13.2.6

The final answer is $\frac{15}{2}$ .

$y = \frac{15}{2}$

Step 14

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

$- 2 \sin (2 (\frac{3 π}{4}))$

Step 15

Evaluate the second derivative.

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Step 15.1

Cancel the common factor of $2$ .

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Step 15.1.1

Factor $2$ out of $4$ .

$- 2 \sin (2 \frac{3 π}{2 (2)})$

Step 15.1.2

Cancel the common factor.

$- 2 \sin (2 \frac{3 π}{2 \cdot 2})$

Step 15.1.3

Rewrite the expression.

$- 2 \sin (\frac{3 π}{2})$

Step 15.2

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.

$- 2 (- \sin (\frac{π}{2}))$

Step 15.3

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

$- 2 (- 1 \cdot 1)$

Step 15.4

Multiply $- 2 (- 1 \cdot 1)$ .

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Step 15.4.1

Multiply $- 1$ by $1$ .

$- 2 \cdot - 1$

Step 15.4.2

Multiply $- 2$ by $- 1$ .

$2$

Step 16

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

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

Step 17

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

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Step 17.1

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

$f (\frac{3 π}{4}) = \sin (\frac{3 π}{4}) \cos (\frac{3 π}{4}) + 7$

Step 17.2

Simplify the result.

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Step 17.2.1

Simplify each term.

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Step 17.2.1.1

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

$f (\frac{3 π}{4}) = \sin (\frac{π}{4}) \cos (\frac{3 π}{4}) + 7$

Step 17.2.1.2

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

$f (\frac{3 π}{4}) = \frac{\sqrt{2}}{2} \cdot \cos (\frac{3 π}{4}) + 7$

Step 17.2.1.3

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.

$f (\frac{3 π}{4}) = \frac{\sqrt{2}}{2} \cdot (- \cos (\frac{π}{4})) + 7$

Step 17.2.1.4

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

$f (\frac{3 π}{4}) = \frac{\sqrt{2}}{2} \cdot (- \frac{\sqrt{2}}{2}) + 7$

Step 17.2.1.5

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

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Step 17.2.1.5.1

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

$f (\frac{3 π}{4}) = - \frac{\sqrt{2} \sqrt{2}}{2 \cdot 2} + 7$

Step 17.2.1.5.2

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

$f (\frac{3 π}{4}) = - \frac{\sqrt{2} \sqrt{2}}{2 \cdot 2} + 7$

Step 17.2.1.5.3

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

$f (\frac{3 π}{4}) = - \frac{\sqrt{2} \sqrt{2}}{2 \cdot 2} + 7$

Step 17.2.1.5.4

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

$f (\frac{3 π}{4}) = - \frac{{\sqrt{2}}^{1 + 1}}{2 \cdot 2} + 7$

Step 17.2.1.5.5

Add $1$ and $1$ .

$f (\frac{3 π}{4}) = - \frac{{\sqrt{2}}^{2}}{2 \cdot 2} + 7$

Step 17.2.1.5.6

Multiply $2$ by $2$ .

$f (\frac{3 π}{4}) = - \frac{{\sqrt{2}}^{2}}{4} + 7$

Step 17.2.1.6

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

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Step 17.2.1.6.1

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

$f (\frac{3 π}{4}) = - \frac{{(2^{\frac{1}{2}})}^{2}}{4} + 7$

Step 17.2.1.6.2

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

$f (\frac{3 π}{4}) = - \frac{2^{\frac{1}{2} \cdot 2}}{4} + 7$

Step 17.2.1.6.3

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

$f (\frac{3 π}{4}) = - \frac{2^{\frac{2}{2}}}{4} + 7$

Step 17.2.1.6.4

Cancel the common factor of $2$ .

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Step 17.2.1.6.4.1

Cancel the common factor.

$f (\frac{3 π}{4}) = - \frac{2^{\frac{2}{2}}}{4} + 7$

Step 17.2.1.6.4.2

Rewrite the expression.

$f (\frac{3 π}{4}) = - \frac{2}{4} + 7$

Step 17.2.1.6.5

Evaluate the exponent.

$f (\frac{3 π}{4}) = - \frac{2}{4} + 7$

Step 17.2.1.7

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

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Step 17.2.1.7.1

Factor $2$ out of $2$ .

$f (\frac{3 π}{4}) = - \frac{2 (1)}{4} + 7$

Step 17.2.1.7.2

Cancel the common factors.

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Step 17.2.1.7.2.1

Factor $2$ out of $4$ .

$f (\frac{3 π}{4}) = - \frac{2 \cdot 1}{2 \cdot 2} + 7$

Step 17.2.1.7.2.2

Cancel the common factor.

$f (\frac{3 π}{4}) = - \frac{2 \cdot 1}{2 \cdot 2} + 7$

Step 17.2.1.7.2.3

Rewrite the expression.

$f (\frac{3 π}{4}) = - \frac{1}{2} + 7$

Step 17.2.2

To write $7$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (\frac{3 π}{4}) = - \frac{1}{2} + 7 \cdot \frac{2}{2}$

Step 17.2.3

Combine $7$ and $\frac{2}{2}$ .

$f (\frac{3 π}{4}) = - \frac{1}{2} + \frac{7 \cdot 2}{2}$

Step 17.2.4

Combine the numerators over the common denominator.

$f (\frac{3 π}{4}) = \frac{- 1 + 7 \cdot 2}{2}$

Step 17.2.5

Simplify the numerator.

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Step 17.2.5.1

Multiply $7$ by $2$ .

$f (\frac{3 π}{4}) = \frac{- 1 + 14}{2}$

Step 17.2.5.2

Add $- 1$ and $14$ .

$f (\frac{3 π}{4}) = \frac{13}{2}$

Step 17.2.6

The final answer is $\frac{13}{2}$ .

$y = \frac{13}{2}$

Step 18

These are the local extrema for $f (x) = \sin (x) \cos (x) + 7$ .

$(\frac{π}{4}, \frac{15}{2})$ is a local maxima

$(\frac{3 π}{4}, \frac{13}{2})$ is a local minima

Step 19