Calculus Examples

$f (x) = 7 \sqrt{2} \cos (x) + 7 \sin^{2} (x)$

Step 1

Find the first derivative of the function.

Tap for more steps...

Step 1.1

By the Sum Rule, the derivative of $7 \sqrt{2} \cos (x) + 7 \sin^{2} (x)$ with respect to $x$ is $\frac{d}{d x} [7 \sqrt{2} \cos (x)] + \frac{d}{d x} [7 \sin^{2} (x)]$ .

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

Step 1.2

Evaluate $\frac{d}{d x} [7 \sqrt{2} \cos (x)]$ .

Tap for more steps...

Step 1.2.1

Since $7 \sqrt{2}$ is constant with respect to $x$ , the derivative of $7 \sqrt{2} \cos (x)$ with respect to $x$ is $7 \sqrt{2} \frac{d}{d x} [\cos (x)]$ .

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

Step 1.2.2

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

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

Step 1.2.3

Multiply $- 1$ by $7$ .

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

Step 1.3

Evaluate $\frac{d}{d x} [7 \sin^{2} (x)]$ .

Tap for more steps...

Step 1.3.1

Since $7$ is constant with respect to $x$ , the derivative of $7 \sin^{2} (x)$ with respect to $x$ is $7 \frac{d}{d x} [\sin^{2} (x)]$ .

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

Step 1.3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = \sin (x)$ .

Tap for more steps...

Step 1.3.2.1

To apply the Chain Rule, set $u$ as $\sin (x)$ .

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

Step 1.3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 2$ .

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

Step 1.3.2.3

Replace all occurrences of $u$ with $\sin (x)$ .

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

Step 1.3.3

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

$- 7 \sqrt{2} \sin (x) + 7 (2 \sin (x) \cos (x))$

Step 1.3.4

Multiply $2$ by $7$ .

$- 7 \sqrt{2} \sin (x) + 14 \sin (x) \cos (x)$

Step 1.4

Reorder terms.

$- 7 \sqrt{2} \sin (x) + 14 \cos (x) \sin (x)$

Step 2

Find the second derivative of the function.

Tap for more steps...

Step 2.1

By the Sum Rule, the derivative of $- 7 \sqrt{2} \sin (x) + 14 \cos (x) \sin (x)$ with respect to $x$ is $\frac{d}{d x} [- 7 \sqrt{2} \sin (x)] + \frac{d}{d x} [14 \cos (x) \sin (x)]$ .

$f'' (x) = \frac{d}{d x} (- 7 \sqrt{2} \sin (x)) + \frac{d}{d x} (14 \cos (x) \sin (x))$

Step 2.2

Evaluate $\frac{d}{d x} [- 7 \sqrt{2} \sin (x)]$ .

Tap for more steps...

Step 2.2.1

Since $- 7 \sqrt{2}$ is constant with respect to $x$ , the derivative of $- 7 \sqrt{2} \sin (x)$ with respect to $x$ is $- 7 \sqrt{2} \frac{d}{d x} [\sin (x)]$ .

$f'' (x) = - 7 \sqrt{2} \frac{d}{d x} \sin (x) + \frac{d}{d x} (14 \cos (x) \sin (x))$

Step 2.2.2

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

$f'' (x) = - 7 \sqrt{2} \cos (x) + \frac{d}{d x} (14 \cos (x) \sin (x))$

Step 2.3

Evaluate $\frac{d}{d x} [14 \cos (x) \sin (x)]$ .

Tap for more steps...

Step 2.3.1

Since $14$ is constant with respect to $x$ , the derivative of $14 \cos (x) \sin (x)$ with respect to $x$ is $14 \frac{d}{d x} [\cos (x) \sin (x)]$ .

$f'' (x) = - 7 \sqrt{2} \cos (x) + 14 \frac{d}{d x} (\cos (x) \sin (x))$

Step 2.3.2

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

$f'' (x) = - 7 \sqrt{2} \cos (x) + 14 (\cos (x) \frac{d}{d x} (\sin (x)) + \sin (x) \frac{d}{d x} (\cos (x)))$

Step 2.3.3

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

$f'' (x) = - 7 \sqrt{2} \cos (x) + 14 (\cos (x) \cos (x) + \sin (x) \frac{d}{d x} (\cos (x)))$

Step 2.3.4

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

$f'' (x) = - 7 \sqrt{2} \cos (x) + 14 (\cos (x) \cos (x) + \sin (x) (- \sin (x)))$

Step 2.3.5

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

$f'' (x) = - 7 \sqrt{2} \cos (x) + 14 (\cos (x) \cos (x) + \sin (x) (- \sin (x)))$

Step 2.3.6

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

$f'' (x) = - 7 \sqrt{2} \cos (x) + 14 (\cos (x) \cos (x) + \sin (x) (- \sin (x)))$

Step 2.3.7

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

$f'' (x) = - 7 \sqrt{2} \cos (x) + 14 ({\cos (x)}^{1 + 1} + \sin (x) (- \sin (x)))$

Step 2.3.8

Add $1$ and $1$ .

$f'' (x) = - 7 \sqrt{2} \cos (x) + 14 (\cos^{2} (x) + \sin (x) (- \sin (x)))$

Step 2.3.9

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

$f'' (x) = - 7 \sqrt{2} \cos (x) + 14 (\cos^{2} (x) - (\sin (x) \sin (x)))$

Step 2.3.10

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

$f'' (x) = - 7 \sqrt{2} \cos (x) + 14 (\cos^{2} (x) - (\sin (x) \sin (x)))$

Step 2.3.11

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

$f'' (x) = - 7 \sqrt{2} \cos (x) + 14 (\cos^{2} (x) - {\sin (x)}^{1 + 1})$

Step 2.3.12

Add $1$ and $1$ .

$f'' (x) = - 7 \sqrt{2} \cos (x) + 14 (\cos^{2} (x) - \sin^{2} (x))$

Step 2.4

Simplify.

Tap for more steps...

Step 2.4.1

Apply the distributive property.

$f'' (x) = - 7 \sqrt{2} \cos (x) + 14 \cos^{2} (x) + 14 (- \sin^{2} (x))$

Step 2.4.2

Multiply $- 1$ by $14$ .

$f'' (x) = - 7 \sqrt{2} \cos (x) + 14 \cos^{2} (x) - 14 \sin^{2} (x)$

Step 3

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

$- 7 \sqrt{2} \sin (x) + 14 \cos (x) \sin (x) = 0$

Step 4

Factor $7 \sin (x)$ out of $- 7 \sqrt{2} \sin (x) + 14 \cos (x) \sin (x)$ .

Tap for more steps...

Step 4.1

Factor $7 \sin (x)$ out of $- 7 \sqrt{2} \sin (x)$ .

$7 \sin (x) (- \sqrt{2}) + 14 \cos (x) \sin (x) = 0$

Step 4.2

Factor $7 \sin (x)$ out of $14 \cos (x) \sin (x)$ .

$7 \sin (x) (- \sqrt{2}) + 7 \sin (x) (2 \cos (x)) = 0$

Step 4.3

Factor $7 \sin (x)$ out of $7 \sin (x) (- \sqrt{2}) + 7 \sin (x) (2 \cos (x))$ .

$7 \sin (x) (- \sqrt{2} + 2 \cos (x)) = 0$

Step 5

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$\sin (x) = 0$

$- \sqrt{2} + 2 \cos (x) = 0$

Step 6

Set $\sin (x)$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 6.1

Set $\sin (x)$ equal to $0$ .

$\sin (x) = 0$

Step 6.2

Solve $\sin (x) = 0$ for $x$ .

Tap for more steps...

Step 6.2.1

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

$x = \arcsin (0)$

Step 6.2.2

Simplify the right side.

Tap for more steps...

Step 6.2.2.1

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

$x = 0$

Step 6.2.3

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from $π$ to find the solution in the second quadrant.

$x = π - 0$

Step 6.2.4

Subtract $0$ from $π$ .

$x = π$

Step 6.2.5

The solution to the equation $x = 0$ .

$x = 0, π$

Step 7

Set $- \sqrt{2} + 2 \cos (x)$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 7.1

Set $- \sqrt{2} + 2 \cos (x)$ equal to $0$ .

$- \sqrt{2} + 2 \cos (x) = 0$

Step 7.2

Solve $- \sqrt{2} + 2 \cos (x) = 0$ for $x$ .

Tap for more steps...

Step 7.2.1

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

$2 \cos (x) = \sqrt{2}$

Step 7.2.2

Divide each term in $2 \cos (x) = \sqrt{2}$ by $2$ and simplify.

Tap for more steps...

Step 7.2.2.1

Divide each term in $2 \cos (x) = \sqrt{2}$ by $2$ .

$\frac{2 \cos (x)}{2} = \frac{\sqrt{2}}{2}$

Step 7.2.2.2

Simplify the left side.

Tap for more steps...

Step 7.2.2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 7.2.2.2.1.1

Cancel the common factor.

$\frac{2 \cos (x)}{2} = \frac{\sqrt{2}}{2}$

Step 7.2.2.2.1.2

Divide $\cos (x)$ by $1$ .

$\cos (x) = \frac{\sqrt{2}}{2}$

Step 7.2.3

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

$x = \arccos (\frac{\sqrt{2}}{2})$

Step 7.2.4

Simplify the right side.

Tap for more steps...

Step 7.2.4.1

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

$x = \frac{π}{4}$

Step 7.2.5

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.

$x = 2 π - \frac{π}{4}$

Step 7.2.6

Simplify $2 π - \frac{π}{4}$ .

Tap for more steps...

Step 7.2.6.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Step 7.2.6.2

Combine fractions.

Tap for more steps...

Step 7.2.6.2.1

Combine $2 π$ and $\frac{4}{4}$ .

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

Step 7.2.6.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 4 - π}{4}$

Step 7.2.6.3

Simplify the numerator.

Tap for more steps...

Step 7.2.6.3.1

Multiply $4$ by $2$ .

$x = \frac{8 π - π}{4}$

Step 7.2.6.3.2

Subtract $π$ from $8 π$ .

$x = \frac{7 π}{4}$

Step 7.2.7

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

$x = \frac{π}{4}, \frac{7 π}{4}$

Step 8

The final solution is all the values that make $7 \sin (x) (- \sqrt{2} + 2 \cos (x)) = 0$ true.

$x = 0, π, \frac{π}{4}, \frac{7 π}{4}$

Step 9

Evaluate the second derivative at $x = 0$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- 7 \sqrt{2} \cos (0) + 14 \cos^{2} (0) - 14 \sin^{2} (0)$

Step 10

Evaluate the second derivative.

Tap for more steps...

Step 10.1

Simplify each term.

Tap for more steps...

Step 10.1.1

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

$- 7 \sqrt{2} \cdot 1 + 14 \cos^{2} (0) - 14 \sin^{2} (0)$

Step 10.1.2

Multiply $- 7$ by $1$ .

$- 7 \sqrt{2} + 14 \cos^{2} (0) - 14 \sin^{2} (0)$

Step 10.1.3

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

$- 7 \sqrt{2} + 14 \cdot 1^{2} - 14 \sin^{2} (0)$

Step 10.1.4

One to any power is one.

$- 7 \sqrt{2} + 14 \cdot 1 - 14 \sin^{2} (0)$

Step 10.1.5

Multiply $14$ by $1$ .

$- 7 \sqrt{2} + 14 - 14 \sin^{2} (0)$

Step 10.1.6

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

$- 7 \sqrt{2} + 14 - 14 \cdot 0^{2}$

Step 10.1.7

Raising $0$ to any positive power yields $0$ .

$- 7 \sqrt{2} + 14 - 14 \cdot 0$

Step 10.1.8

Multiply $- 14$ by $0$ .

$- 7 \sqrt{2} + 14 + 0$

Step 10.2

Add $- 7 \sqrt{2}$ and $0$ .

$14 - 7 \sqrt{2}$

Step 11

$x = 0$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = 0$ is a local minimum

Step 12

Find the y-value when $x = 0$ .

Tap for more steps...

Step 12.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = 7 \sqrt{2} \cos (0) + 7 \sin^{2} (0)$

Step 12.2

Simplify the result.

Tap for more steps...

Step 12.2.1

Simplify each term.

Tap for more steps...

Step 12.2.1.1

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

$f (0) = 7 \sqrt{2} \cdot 1 + 7 \sin^{2} (0)$

Step 12.2.1.2

Multiply $7$ by $1$ .

$f (0) = 7 \sqrt{2} + 7 \sin^{2} (0)$

Step 12.2.1.3

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

$f (0) = 7 \sqrt{2} + 7 \cdot 0^{2}$

Step 12.2.1.4

Raising $0$ to any positive power yields $0$ .

$f (0) = 7 \sqrt{2} + 7 \cdot 0$

Step 12.2.1.5

Multiply $7$ by $0$ .

$f (0) = 7 \sqrt{2} + 0$

Step 12.2.2

Add $7 \sqrt{2}$ and $0$ .

$f (0) = 7 \sqrt{2}$

Step 12.2.3

The final answer is $7 \sqrt{2}$ .

$y = 7 \sqrt{2}$

Step 13

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.

$- 7 \sqrt{2} \cos (π) + 14 \cos^{2} (π) - 14 \sin^{2} (π)$

Step 14

Evaluate the second derivative.

Tap for more steps...

Step 14.1

Simplify each term.

Tap for more steps...

Step 14.1.1

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.

$- 7 \sqrt{2} (- \cos (0)) + 14 \cos^{2} (π) - 14 \sin^{2} (π)$

Step 14.1.2

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

$- 7 \sqrt{2} (- 1 \cdot 1) + 14 \cos^{2} (π) - 14 \sin^{2} (π)$

Step 14.1.3

Multiply $- 1$ by $1$ .

$- 7 \sqrt{2} \cdot - 1 + 14 \cos^{2} (π) - 14 \sin^{2} (π)$

Step 14.1.4

Multiply $- 1$ by $- 7$ .

$7 \sqrt{2} + 14 \cos^{2} (π) - 14 \sin^{2} (π)$

Step 14.1.5

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.

$7 \sqrt{2} + 14 {(- \cos (0))}^{2} - 14 \sin^{2} (π)$

Step 14.1.6

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

$7 \sqrt{2} + 14 {(- 1 \cdot 1)}^{2} - 14 \sin^{2} (π)$

Step 14.1.7

Multiply $- 1$ by $1$ .

$7 \sqrt{2} + 14 {(- 1)}^{2} - 14 \sin^{2} (π)$

Step 14.1.8

Raise $- 1$ to the power of $2$ .

$7 \sqrt{2} + 14 \cdot 1 - 14 \sin^{2} (π)$

Step 14.1.9

Multiply $14$ by $1$ .

$7 \sqrt{2} + 14 - 14 \sin^{2} (π)$

Step 14.1.10

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

$7 \sqrt{2} + 14 - 14 \sin^{2} (0)$

Step 14.1.11

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

$7 \sqrt{2} + 14 - 14 \cdot 0^{2}$

Step 14.1.12

Raising $0$ to any positive power yields $0$ .

$7 \sqrt{2} + 14 - 14 \cdot 0$

Step 14.1.13

Multiply $- 14$ by $0$ .

$7 \sqrt{2} + 14 + 0$

Step 14.2

Add $7 \sqrt{2}$ and $0$ .

$14 + 7 \sqrt{2}$

Step 15

$x = π$ 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 16

Find the y-value when $x = π$ .

Tap for more steps...

Step 16.1

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

$f (π) = 7 \sqrt{2} \cos (π) + 7 \sin^{2} (π)$

Step 16.2

Simplify the result.

Tap for more steps...

Step 16.2.1

Simplify each term.

Tap for more steps...

Step 16.2.1.1

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.

$f (π) = 7 \sqrt{2} (- \cos (0)) + 7 \sin^{2} (π)$

Step 16.2.1.2

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

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

Step 16.2.1.3

Multiply $- 1$ by $1$ .

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

Step 16.2.1.4

Multiply $- 1$ by $7$ .

$f (π) = - 7 \sqrt{2} + 7 \sin^{2} (π)$

Step 16.2.1.5

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

$f (π) = - 7 \sqrt{2} + 7 \sin^{2} (0)$

Step 16.2.1.6

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

$f (π) = - 7 \sqrt{2} + 7 \cdot 0^{2}$

Step 16.2.1.7

Raising $0$ to any positive power yields $0$ .

$f (π) = - 7 \sqrt{2} + 7 \cdot 0$

Step 16.2.1.8

Multiply $7$ by $0$ .

$f (π) = - 7 \sqrt{2} + 0$

Step 16.2.2

Add $- 7 \sqrt{2}$ and $0$ .

$f (π) = - 7 \sqrt{2}$

Step 16.2.3

The final answer is $- 7 \sqrt{2}$ .

$y = - 7 \sqrt{2}$

Step 17

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.

$- 7 \sqrt{2} \cos (\frac{π}{4}) + 14 \cos^{2} (\frac{π}{4}) - 14 \sin^{2} (\frac{π}{4})$

Step 18

Evaluate the second derivative.

Tap for more steps...

Step 18.1

Simplify each term.

Tap for more steps...

Step 18.1.1

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

$- 7 \sqrt{2} \frac{\sqrt{2}}{2} + 14 \cos^{2} (\frac{π}{4}) - 14 \sin^{2} (\frac{π}{4})$

Step 18.1.2

Multiply $- 7 \sqrt{2} \frac{\sqrt{2}}{2}$ .

Tap for more steps...

Step 18.1.2.1

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

$\frac{\sqrt{2} \cdot - 7}{2} \sqrt{2} + 14 \cos^{2} (\frac{π}{4}) - 14 \sin^{2} (\frac{π}{4})$

Step 18.1.2.2

Combine $\frac{\sqrt{2} \cdot - 7}{2}$ and $\sqrt{2}$ .

$\frac{\sqrt{2} \cdot - 7 \sqrt{2}}{2} + 14 \cos^{2} (\frac{π}{4}) - 14 \sin^{2} (\frac{π}{4})$

Step 18.1.2.3

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

$\frac{- 7 ({\sqrt{2}}^{1} \sqrt{2})}{2} + 14 \cos^{2} (\frac{π}{4}) - 14 \sin^{2} (\frac{π}{4})$

Step 18.1.2.4

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

$\frac{- 7 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}{2} + 14 \cos^{2} (\frac{π}{4}) - 14 \sin^{2} (\frac{π}{4})$

Step 18.1.2.5

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

$\frac{- 7 {\sqrt{2}}^{1 + 1}}{2} + 14 \cos^{2} (\frac{π}{4}) - 14 \sin^{2} (\frac{π}{4})$

Step 18.1.2.6

Add $1$ and $1$ .

$\frac{- 7 {\sqrt{2}}^{2}}{2} + 14 \cos^{2} (\frac{π}{4}) - 14 \sin^{2} (\frac{π}{4})$

Step 18.1.3

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

Tap for more steps...

Step 18.1.3.1

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

$\frac{- 7 {(2^{\frac{1}{2}})}^{2}}{2} + 14 \cos^{2} (\frac{π}{4}) - 14 \sin^{2} (\frac{π}{4})$

Step 18.1.3.2

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

$\frac{- 7 \cdot 2^{\frac{1}{2} \cdot 2}}{2} + 14 \cos^{2} (\frac{π}{4}) - 14 \sin^{2} (\frac{π}{4})$

Step 18.1.3.3

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

$\frac{- 7 \cdot 2^{\frac{2}{2}}}{2} + 14 \cos^{2} (\frac{π}{4}) - 14 \sin^{2} (\frac{π}{4})$

Step 18.1.3.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 18.1.3.4.1

Cancel the common factor.

$\frac{- 7 \cdot 2^{\frac{2}{2}}}{2} + 14 \cos^{2} (\frac{π}{4}) - 14 \sin^{2} (\frac{π}{4})$

Step 18.1.3.4.2

Rewrite the expression.

$\frac{- 7 \cdot 2^{1}}{2} + 14 \cos^{2} (\frac{π}{4}) - 14 \sin^{2} (\frac{π}{4})$

Step 18.1.3.5

Evaluate the exponent.

$\frac{- 7 \cdot 2}{2} + 14 \cos^{2} (\frac{π}{4}) - 14 \sin^{2} (\frac{π}{4})$

Step 18.1.4

Multiply $- 7$ by $2$ .

$\frac{- 14}{2} + 14 \cos^{2} (\frac{π}{4}) - 14 \sin^{2} (\frac{π}{4})$

Step 18.1.5

Divide $- 14$ by $2$ .

$- 7 + 14 \cos^{2} (\frac{π}{4}) - 14 \sin^{2} (\frac{π}{4})$

Step 18.1.6

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

$- 7 + 14 {(\frac{\sqrt{2}}{2})}^{2} - 14 \sin^{2} (\frac{π}{4})$

Step 18.1.7

Apply the product rule to $\frac{\sqrt{2}}{2}$ .

$- 7 + 14 \frac{{\sqrt{2}}^{2}}{2^{2}} - 14 \sin^{2} (\frac{π}{4})$

Step 18.1.8

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

Tap for more steps...

Step 18.1.8.1

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

$- 7 + 14 \frac{{(2^{\frac{1}{2}})}^{2}}{2^{2}} - 14 \sin^{2} (\frac{π}{4})$

Step 18.1.8.2

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

$- 7 + 14 \frac{2^{\frac{1}{2} \cdot 2}}{2^{2}} - 14 \sin^{2} (\frac{π}{4})$

Step 18.1.8.3

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

$- 7 + 14 \frac{2^{\frac{2}{2}}}{2^{2}} - 14 \sin^{2} (\frac{π}{4})$

Step 18.1.8.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 18.1.8.4.1

Cancel the common factor.

$- 7 + 14 \frac{2^{\frac{2}{2}}}{2^{2}} - 14 \sin^{2} (\frac{π}{4})$

Step 18.1.8.4.2

Rewrite the expression.

$- 7 + 14 \frac{2^{1}}{2^{2}} - 14 \sin^{2} (\frac{π}{4})$

Step 18.1.8.5

Evaluate the exponent.

$- 7 + 14 \frac{2}{2^{2}} - 14 \sin^{2} (\frac{π}{4})$

Step 18.1.9

Raise $2$ to the power of $2$ .

$- 7 + 14 (\frac{2}{4}) - 14 \sin^{2} (\frac{π}{4})$

Step 18.1.10

Cancel the common factor of $2$ .

Tap for more steps...

Step 18.1.10.1

Factor $2$ out of $14$ .

$- 7 + 2 (7) \frac{2}{4} - 14 \sin^{2} (\frac{π}{4})$

Step 18.1.10.2

Factor $2$ out of $4$ .

$- 7 + 2 \cdot 7 \frac{2}{2 \cdot 2} - 14 \sin^{2} (\frac{π}{4})$

Step 18.1.10.3

Cancel the common factor.

$- 7 + 2 \cdot 7 \frac{2}{2 \cdot 2} - 14 \sin^{2} (\frac{π}{4})$

Step 18.1.10.4

Rewrite the expression.

$- 7 + 7 (\frac{2}{2}) - 14 \sin^{2} (\frac{π}{4})$

Step 18.1.11

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

$- 7 + \frac{7 \cdot 2}{2} - 14 \sin^{2} (\frac{π}{4})$

Step 18.1.12

Multiply $7$ by $2$ .

$- 7 + \frac{14}{2} - 14 \sin^{2} (\frac{π}{4})$

Step 18.1.13

Divide $14$ by $2$ .

$- 7 + 7 - 14 \sin^{2} (\frac{π}{4})$

Step 18.1.14

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

$- 7 + 7 - 14 {(\frac{\sqrt{2}}{2})}^{2}$

Step 18.1.15

Apply the product rule to $\frac{\sqrt{2}}{2}$ .

$- 7 + 7 - 14 \frac{{\sqrt{2}}^{2}}{2^{2}}$

Step 18.1.16

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

Tap for more steps...

Step 18.1.16.1

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

$- 7 + 7 - 14 \frac{{(2^{\frac{1}{2}})}^{2}}{2^{2}}$

Step 18.1.16.2

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

$- 7 + 7 - 14 \frac{2^{\frac{1}{2} \cdot 2}}{2^{2}}$

Step 18.1.16.3

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

$- 7 + 7 - 14 \frac{2^{\frac{2}{2}}}{2^{2}}$

Step 18.1.16.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 18.1.16.4.1

Cancel the common factor.

$- 7 + 7 - 14 \frac{2^{\frac{2}{2}}}{2^{2}}$

Step 18.1.16.4.2

Rewrite the expression.

$- 7 + 7 - 14 \frac{2^{1}}{2^{2}}$

Step 18.1.16.5

Evaluate the exponent.

$- 7 + 7 - 14 \frac{2}{2^{2}}$

Step 18.1.17

Raise $2$ to the power of $2$ .

$- 7 + 7 - 14 (\frac{2}{4})$

Step 18.1.18

Cancel the common factor of $2$ .

Tap for more steps...

Step 18.1.18.1

Factor $2$ out of $- 14$ .

$- 7 + 7 + 2 (- 7) \frac{2}{4}$

Step 18.1.18.2

Factor $2$ out of $4$ .

$- 7 + 7 + 2 \cdot - 7 \frac{2}{2 \cdot 2}$

Step 18.1.18.3

Cancel the common factor.

$- 7 + 7 + 2 \cdot - 7 \frac{2}{2 \cdot 2}$

Step 18.1.18.4

Rewrite the expression.

$- 7 + 7 - 7 (\frac{2}{2})$

Step 18.1.19

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

$- 7 + 7 + \frac{- 7 \cdot 2}{2}$

Step 18.1.20

Multiply $- 7$ by $2$ .

$- 7 + 7 + \frac{- 14}{2}$

Step 18.1.21

Divide $- 14$ by $2$ .

$- 7 + 7 - 7$

Step 18.2

Simplify by adding and subtracting.

Tap for more steps...

Step 18.2.1

Add $- 7$ and $7$ .

$0 - 7$

Step 18.2.2

Subtract $7$ from $0$ .

$- 7$

Step 19

$x = \frac{π}{4}$ 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 20

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

Tap for more steps...

Step 20.1

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

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

Step 20.2

Simplify the result.

Tap for more steps...

Step 20.2.1

Simplify each term.

Tap for more steps...

Step 20.2.1.1

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

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

Step 20.2.1.2

Multiply $7 \sqrt{2} \frac{\sqrt{2}}{2}$ .

Tap for more steps...

Step 20.2.1.2.1

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

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

Step 20.2.1.2.2

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

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

Step 20.2.1.2.3

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

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

Step 20.2.1.2.4

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

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

Step 20.2.1.2.5

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

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

Step 20.2.1.2.6

Add $1$ and $1$ .

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

Step 20.2.1.3

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

Tap for more steps...

Step 20.2.1.3.1

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

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

Step 20.2.1.3.2

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

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

Step 20.2.1.3.3

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

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

Step 20.2.1.3.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 20.2.1.3.4.1

Cancel the common factor.

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

Step 20.2.1.3.4.2

Rewrite the expression.

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

Step 20.2.1.3.5

Evaluate the exponent.

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

Step 20.2.1.4

Multiply $7$ by $2$ .

$f (\frac{π}{4}) = \frac{14}{2} + 7 \sin^{2} (\frac{π}{4})$

Step 20.2.1.5

Divide $14$ by $2$ .

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

Step 20.2.1.6

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

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

Step 20.2.1.7

Apply the product rule to $\frac{\sqrt{2}}{2}$ .

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

Step 20.2.1.8

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

Tap for more steps...

Step 20.2.1.8.1

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

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

Step 20.2.1.8.2

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

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

Step 20.2.1.8.3

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

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

Step 20.2.1.8.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 20.2.1.8.4.1

Cancel the common factor.

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

Step 20.2.1.8.4.2

Rewrite the expression.

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

Step 20.2.1.8.5

Evaluate the exponent.

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

Step 20.2.1.9

Raise $2$ to the power of $2$ .

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

Step 20.2.1.10

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

Tap for more steps...

Step 20.2.1.10.1

Factor $2$ out of $2$ .

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

Step 20.2.1.10.2

Cancel the common factors.

Tap for more steps...

Step 20.2.1.10.2.1

Factor $2$ out of $4$ .

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

Step 20.2.1.10.2.2

Cancel the common factor.

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

Step 20.2.1.10.2.3

Rewrite the expression.

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

Step 20.2.1.11

Combine $7$ and $\frac{1}{2}$ .

$f (\frac{π}{4}) = 7 + \frac{7}{2}$

Step 20.2.2

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

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

Step 20.2.3

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

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

Step 20.2.4

Combine the numerators over the common denominator.

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

Step 20.2.5

Simplify the numerator.

Tap for more steps...

Step 20.2.5.1

Multiply $7$ by $2$ .

$f (\frac{π}{4}) = \frac{14 + 7}{2}$

Step 20.2.5.2

Add $14$ and $7$ .

$f (\frac{π}{4}) = \frac{21}{2}$

Step 20.2.6

The final answer is $\frac{21}{2}$ .

$y = \frac{21}{2}$

Step 21

Evaluate the second derivative at $x = \frac{7 π}{4}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- 7 \sqrt{2} \cos (\frac{7 π}{4}) + 14 \cos^{2} (\frac{7 π}{4}) - 14 \sin^{2} (\frac{7 π}{4})$

Step 22

Evaluate the second derivative.

Tap for more steps...

Step 22.1

Simplify each term.

Tap for more steps...

Step 22.1.1

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

$- 7 \sqrt{2} \cos (\frac{π}{4}) + 14 \cos^{2} (\frac{7 π}{4}) - 14 \sin^{2} (\frac{7 π}{4})$

Step 22.1.2

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

$- 7 \sqrt{2} \frac{\sqrt{2}}{2} + 14 \cos^{2} (\frac{7 π}{4}) - 14 \sin^{2} (\frac{7 π}{4})$

Step 22.1.3

Multiply $- 7 \sqrt{2} \frac{\sqrt{2}}{2}$ .

Tap for more steps...

Step 22.1.3.1

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

$\frac{\sqrt{2} \cdot - 7}{2} \sqrt{2} + 14 \cos^{2} (\frac{7 π}{4}) - 14 \sin^{2} (\frac{7 π}{4})$

Step 22.1.3.2

Combine $\frac{\sqrt{2} \cdot - 7}{2}$ and $\sqrt{2}$ .

$\frac{\sqrt{2} \cdot - 7 \sqrt{2}}{2} + 14 \cos^{2} (\frac{7 π}{4}) - 14 \sin^{2} (\frac{7 π}{4})$

Step 22.1.3.3

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

$\frac{- 7 ({\sqrt{2}}^{1} \sqrt{2})}{2} + 14 \cos^{2} (\frac{7 π}{4}) - 14 \sin^{2} (\frac{7 π}{4})$

Step 22.1.3.4

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

$\frac{- 7 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}{2} + 14 \cos^{2} (\frac{7 π}{4}) - 14 \sin^{2} (\frac{7 π}{4})$

Step 22.1.3.5

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

$\frac{- 7 {\sqrt{2}}^{1 + 1}}{2} + 14 \cos^{2} (\frac{7 π}{4}) - 14 \sin^{2} (\frac{7 π}{4})$

Step 22.1.3.6

Add $1$ and $1$ .

$\frac{- 7 {\sqrt{2}}^{2}}{2} + 14 \cos^{2} (\frac{7 π}{4}) - 14 \sin^{2} (\frac{7 π}{4})$

Step 22.1.4

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

Tap for more steps...

Step 22.1.4.1

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

$\frac{- 7 {(2^{\frac{1}{2}})}^{2}}{2} + 14 \cos^{2} (\frac{7 π}{4}) - 14 \sin^{2} (\frac{7 π}{4})$

Step 22.1.4.2

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

$\frac{- 7 \cdot 2^{\frac{1}{2} \cdot 2}}{2} + 14 \cos^{2} (\frac{7 π}{4}) - 14 \sin^{2} (\frac{7 π}{4})$

Step 22.1.4.3

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

$\frac{- 7 \cdot 2^{\frac{2}{2}}}{2} + 14 \cos^{2} (\frac{7 π}{4}) - 14 \sin^{2} (\frac{7 π}{4})$

Step 22.1.4.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 22.1.4.4.1

Cancel the common factor.

$\frac{- 7 \cdot 2^{\frac{2}{2}}}{2} + 14 \cos^{2} (\frac{7 π}{4}) - 14 \sin^{2} (\frac{7 π}{4})$

Step 22.1.4.4.2

Rewrite the expression.

$\frac{- 7 \cdot 2^{1}}{2} + 14 \cos^{2} (\frac{7 π}{4}) - 14 \sin^{2} (\frac{7 π}{4})$

Step 22.1.4.5

Evaluate the exponent.

$\frac{- 7 \cdot 2}{2} + 14 \cos^{2} (\frac{7 π}{4}) - 14 \sin^{2} (\frac{7 π}{4})$

Step 22.1.5

Multiply $- 7$ by $2$ .

$\frac{- 14}{2} + 14 \cos^{2} (\frac{7 π}{4}) - 14 \sin^{2} (\frac{7 π}{4})$

Step 22.1.6

Divide $- 14$ by $2$ .

$- 7 + 14 \cos^{2} (\frac{7 π}{4}) - 14 \sin^{2} (\frac{7 π}{4})$

Step 22.1.7

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

$- 7 + 14 \cos^{2} (\frac{π}{4}) - 14 \sin^{2} (\frac{7 π}{4})$

Step 22.1.8

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

$- 7 + 14 {(\frac{\sqrt{2}}{2})}^{2} - 14 \sin^{2} (\frac{7 π}{4})$

Step 22.1.9

Apply the product rule to $\frac{\sqrt{2}}{2}$ .

$- 7 + 14 \frac{{\sqrt{2}}^{2}}{2^{2}} - 14 \sin^{2} (\frac{7 π}{4})$

Step 22.1.10

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

Tap for more steps...

Step 22.1.10.1

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

$- 7 + 14 \frac{{(2^{\frac{1}{2}})}^{2}}{2^{2}} - 14 \sin^{2} (\frac{7 π}{4})$

Step 22.1.10.2

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

$- 7 + 14 \frac{2^{\frac{1}{2} \cdot 2}}{2^{2}} - 14 \sin^{2} (\frac{7 π}{4})$

Step 22.1.10.3

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

$- 7 + 14 \frac{2^{\frac{2}{2}}}{2^{2}} - 14 \sin^{2} (\frac{7 π}{4})$

Step 22.1.10.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 22.1.10.4.1

Cancel the common factor.

$- 7 + 14 \frac{2^{\frac{2}{2}}}{2^{2}} - 14 \sin^{2} (\frac{7 π}{4})$

Step 22.1.10.4.2

Rewrite the expression.

$- 7 + 14 \frac{2^{1}}{2^{2}} - 14 \sin^{2} (\frac{7 π}{4})$

Step 22.1.10.5

Evaluate the exponent.

$- 7 + 14 \frac{2}{2^{2}} - 14 \sin^{2} (\frac{7 π}{4})$

Step 22.1.11

Raise $2$ to the power of $2$ .

$- 7 + 14 (\frac{2}{4}) - 14 \sin^{2} (\frac{7 π}{4})$

Step 22.1.12

Cancel the common factor of $2$ .

Tap for more steps...

Step 22.1.12.1

Factor $2$ out of $14$ .

$- 7 + 2 (7) \frac{2}{4} - 14 \sin^{2} (\frac{7 π}{4})$

Step 22.1.12.2

Factor $2$ out of $4$ .

$- 7 + 2 \cdot 7 \frac{2}{2 \cdot 2} - 14 \sin^{2} (\frac{7 π}{4})$

Step 22.1.12.3

Cancel the common factor.

$- 7 + 2 \cdot 7 \frac{2}{2 \cdot 2} - 14 \sin^{2} (\frac{7 π}{4})$

Step 22.1.12.4

Rewrite the expression.

$- 7 + 7 (\frac{2}{2}) - 14 \sin^{2} (\frac{7 π}{4})$

Step 22.1.13

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

$- 7 + \frac{7 \cdot 2}{2} - 14 \sin^{2} (\frac{7 π}{4})$

Step 22.1.14

Multiply $7$ by $2$ .

$- 7 + \frac{14}{2} - 14 \sin^{2} (\frac{7 π}{4})$

Step 22.1.15

Divide $14$ by $2$ .

$- 7 + 7 - 14 \sin^{2} (\frac{7 π}{4})$

Step 22.1.16

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.

$- 7 + 7 - 14 {(- \sin (\frac{π}{4}))}^{2}$

Step 22.1.17

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

$- 7 + 7 - 14 {(- \frac{\sqrt{2}}{2})}^{2}$

Step 22.1.18

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 22.1.18.1

Apply the product rule to $- \frac{\sqrt{2}}{2}$ .

$- 7 + 7 - 14 ({(- 1)}^{2} {(\frac{\sqrt{2}}{2})}^{2})$

Step 22.1.18.2

Apply the product rule to $\frac{\sqrt{2}}{2}$ .

$- 7 + 7 - 14 ({(- 1)}^{2} \frac{{\sqrt{2}}^{2}}{2^{2}})$

Step 22.1.19

Raise $- 1$ to the power of $2$ .

$- 7 + 7 - 14 (1 \frac{{\sqrt{2}}^{2}}{2^{2}})$

Step 22.1.20

Multiply $\frac{{\sqrt{2}}^{2}}{2^{2}}$ by $1$ .

$- 7 + 7 - 14 \frac{{\sqrt{2}}^{2}}{2^{2}}$

Step 22.1.21

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

Tap for more steps...

Step 22.1.21.1

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

$- 7 + 7 - 14 \frac{{(2^{\frac{1}{2}})}^{2}}{2^{2}}$

Step 22.1.21.2

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

$- 7 + 7 - 14 \frac{2^{\frac{1}{2} \cdot 2}}{2^{2}}$

Step 22.1.21.3

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

$- 7 + 7 - 14 \frac{2^{\frac{2}{2}}}{2^{2}}$

Step 22.1.21.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 22.1.21.4.1

Cancel the common factor.

$- 7 + 7 - 14 \frac{2^{\frac{2}{2}}}{2^{2}}$

Step 22.1.21.4.2

Rewrite the expression.

$- 7 + 7 - 14 \frac{2^{1}}{2^{2}}$

Step 22.1.21.5

Evaluate the exponent.

$- 7 + 7 - 14 \frac{2}{2^{2}}$

Step 22.1.22

Raise $2$ to the power of $2$ .

$- 7 + 7 - 14 (\frac{2}{4})$

Step 22.1.23

Cancel the common factor of $2$ .

Tap for more steps...

Step 22.1.23.1

Factor $2$ out of $- 14$ .

$- 7 + 7 + 2 (- 7) \frac{2}{4}$

Step 22.1.23.2

Factor $2$ out of $4$ .

$- 7 + 7 + 2 \cdot - 7 \frac{2}{2 \cdot 2}$

Step 22.1.23.3

Cancel the common factor.

$- 7 + 7 + 2 \cdot - 7 \frac{2}{2 \cdot 2}$

Step 22.1.23.4

Rewrite the expression.

$- 7 + 7 - 7 (\frac{2}{2})$

Step 22.1.24

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

$- 7 + 7 + \frac{- 7 \cdot 2}{2}$

Step 22.1.25

Multiply $- 7$ by $2$ .

$- 7 + 7 + \frac{- 14}{2}$

Step 22.1.26

Divide $- 14$ by $2$ .

$- 7 + 7 - 7$

Step 22.2

Simplify by adding and subtracting.

Tap for more steps...

Step 22.2.1

Add $- 7$ and $7$ .

$0 - 7$

Step 22.2.2

Subtract $7$ from $0$ .

$- 7$

Step 23

$x = \frac{7 π}{4}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = \frac{7 π}{4}$ is a local maximum

Step 24

Find the y-value when $x = \frac{7 π}{4}$ .

Tap for more steps...

Step 24.1

Replace the variable $x$ with $\frac{7 π}{4}$ in the expression.

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

Step 24.2

Simplify the result.

Tap for more steps...

Step 24.2.1

Simplify each term.

Tap for more steps...

Step 24.2.1.1

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

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

Step 24.2.1.2

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

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

Step 24.2.1.3

Multiply $7 \sqrt{2} \frac{\sqrt{2}}{2}$ .

Tap for more steps...

Step 24.2.1.3.1

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

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

Step 24.2.1.3.2

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

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

Step 24.2.1.3.3

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

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

Step 24.2.1.3.4

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

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

Step 24.2.1.3.5

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

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

Step 24.2.1.3.6

Add $1$ and $1$ .

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

Step 24.2.1.4

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

Tap for more steps...

Step 24.2.1.4.1

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

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

Step 24.2.1.4.2

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

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

Step 24.2.1.4.3

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

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

Step 24.2.1.4.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 24.2.1.4.4.1

Cancel the common factor.

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

Step 24.2.1.4.4.2

Rewrite the expression.

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

Step 24.2.1.4.5

Evaluate the exponent.

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

Step 24.2.1.5

Multiply $7$ by $2$ .

$f (\frac{7 π}{4}) = \frac{14}{2} + 7 \sin^{2} (\frac{7 π}{4})$

Step 24.2.1.6

Divide $14$ by $2$ .

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

Step 24.2.1.7

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.

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

Step 24.2.1.8

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

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

Step 24.2.1.9

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 24.2.1.9.1

Apply the product rule to $- \frac{\sqrt{2}}{2}$ .

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

Step 24.2.1.9.2

Apply the product rule to $\frac{\sqrt{2}}{2}$ .

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

Step 24.2.1.10

Raise $- 1$ to the power of $2$ .

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

Step 24.2.1.11

Multiply $\frac{{\sqrt{2}}^{2}}{2^{2}}$ by $1$ .

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

Step 24.2.1.12

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

Tap for more steps...

Step 24.2.1.12.1

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

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

Step 24.2.1.12.2

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

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

Step 24.2.1.12.3

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

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

Step 24.2.1.12.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 24.2.1.12.4.1

Cancel the common factor.

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

Step 24.2.1.12.4.2

Rewrite the expression.

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

Step 24.2.1.12.5

Evaluate the exponent.

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

Step 24.2.1.13

Raise $2$ to the power of $2$ .

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

Step 24.2.1.14

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

Tap for more steps...

Step 24.2.1.14.1

Factor $2$ out of $2$ .

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

Step 24.2.1.14.2

Cancel the common factors.

Tap for more steps...

Step 24.2.1.14.2.1

Factor $2$ out of $4$ .

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

Step 24.2.1.14.2.2

Cancel the common factor.

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

Step 24.2.1.14.2.3

Rewrite the expression.

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

Step 24.2.1.15

Combine $7$ and $\frac{1}{2}$ .

$f (\frac{7 π}{4}) = 7 + \frac{7}{2}$

Step 24.2.2

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

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

Step 24.2.3

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

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

Step 24.2.4

Combine the numerators over the common denominator.

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

Step 24.2.5

Simplify the numerator.

Tap for more steps...

Step 24.2.5.1

Multiply $7$ by $2$ .

$f (\frac{7 π}{4}) = \frac{14 + 7}{2}$

Step 24.2.5.2

Add $14$ and $7$ .

$f (\frac{7 π}{4}) = \frac{21}{2}$

Step 24.2.6

The final answer is $\frac{21}{2}$ .

$y = \frac{21}{2}$

Step 25

These are the local extrema for $f (x) = 7 \sqrt{2} \cos (x) + 7 \sin^{2} (x)$ .

$(0, 7 \sqrt{2})$ is a local minima

$(π, - 7 \sqrt{2})$ is a local minima

$(\frac{π}{4}, \frac{21}{2})$ is a local maxima

$(\frac{7 π}{4}, \frac{21}{2})$ is a local maxima

Step 26