Precalculus Examples

$y = - \cos (8 x)$

Step 1

Find the first derivative of the function.

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Since $- 1$ is constant with respect to $x$ , the derivative of $- \cos (8 x)$ with respect to $x$ is $- \frac{d}{d x} [\cos (8 x)]$ .

$- \frac{d}{d x} [\cos (8 x)]$

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 $g (x) = 8 x$ .

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To apply the Chain Rule, set $u$ as $8 x$ .

$- (\frac{d}{d u} [\cos (u)] \frac{d}{d x} [8 x])$

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

$- (- \sin (u) \frac{d}{d x} [8 x])$

Replace all occurrences of $u$ with $8 x$ .

$- (- \sin (8 x) \frac{d}{d x} [8 x])$

Differentiate.

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Multiply $- 1$ by $- 1$ .

$1 (\sin (8 x) \frac{d}{d x} [8 x])$

Multiply $\sin (8 x)$ by $1$ .

$\sin (8 x) \frac{d}{d x} [8 x]$

Since $8$ is constant with respect to $x$ , the derivative of $8 x$ with respect to $x$ is $8 \frac{d}{d x} [x]$ .

$\sin (8 x) (8 \frac{d}{d x} [x])$

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

$\sin (8 x) (8 \cdot 1)$

Simplify the expression.

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Multiply $8$ by $1$ .

$\sin (8 x) \cdot 8$

Move $8$ to the left of $\sin (8 x)$ .

$8 \sin (8 x)$

Step 2

Find the second derivative of the function.

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Since $8$ is constant with respect to $x$ , the derivative of $8 \sin (8 x)$ with respect to $x$ is $8 \frac{d}{d x} [\sin (8 x)]$ .

$f'' (x) = 8 \frac{d}{d x} (\sin (8 x))$

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 $g (x) = 8 x$ .

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To apply the Chain Rule, set $u$ as $8 x$ .

$f'' (x) = 8 (\frac{d}{d u} (\sin (u)) \frac{d}{d x} (8 x))$

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

$f'' (x) = 8 (\cos (u) \frac{d}{d x} (8 x))$

Replace all occurrences of $u$ with $8 x$ .

$f'' (x) = 8 (\cos (8 x) \frac{d}{d x} (8 x))$

Differentiate.

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Since $8$ is constant with respect to $x$ , the derivative of $8 x$ with respect to $x$ is $8 \frac{d}{d x} [x]$ .

$f'' (x) = 8 \cos (8 x) (8 \frac{d}{d x} (x))$

Multiply $8$ by $8$ .

$f'' (x) = 64 \cos (8 x) \frac{d}{d x} (x)$

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

$f'' (x) = 64 \cos (8 x) \cdot 1$

Multiply $64$ by $1$ .

$f'' (x) = 64 \cos (8 x)$

Step 3

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

$8 \sin (8 x) = 0$

Step 4

Divide each term in $8 \sin (8 x) = 0$ by $8$ and simplify.

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Divide each term in $8 \sin (8 x) = 0$ by $8$ .

$\frac{8 \sin (8 x)}{8} = \frac{0}{8}$

Simplify the left side.

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Cancel the common factor of $8$ .

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Cancel the common factor.

$\frac{8 \sin (8 x)}{8} = \frac{0}{8}$

Divide $\sin (8 x)$ by $1$ .

$\sin (8 x) = \frac{0}{8}$

Simplify the right side.

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Divide $0$ by $8$ .

$\sin (8 x) = 0$

Step 5

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

$8 x = \arcsin (0)$

Step 6

Simplify the right side.

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

$8 x = 0$

Step 7

Divide each term in $8 x = 0$ by $8$ and simplify.

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Divide each term in $8 x = 0$ by $8$ .

$\frac{8 x}{8} = \frac{0}{8}$

Simplify the left side.

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Cancel the common factor of $8$ .

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Cancel the common factor.

$\frac{8 x}{8} = \frac{0}{8}$

Divide $x$ by $1$ .

$x = \frac{0}{8}$

Simplify the right side.

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Divide $0$ by $8$ .

$x = 0$

Step 8

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.

$8 x = π - 0$

Step 9

Solve for $x$ .

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Simplify.

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Multiply $- 1$ by $0$ .

$8 x = π + 0$

Add $π$ and $0$ .

$8 x = π$

Divide each term in $8 x = π$ by $8$ and simplify.

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Divide each term in $8 x = π$ by $8$ .

$\frac{8 x}{8} = \frac{π}{8}$

Simplify the left side.

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Cancel the common factor of $8$ .

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Cancel the common factor.

$\frac{8 x}{8} = \frac{π}{8}$

Divide $x$ by $1$ .

$x = \frac{π}{8}$

Step 10

The solution to the equation $8 x = 0$ .

$x = 0, \frac{π}{8}$

Step 11

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.

$64 \cos (8 (0))$

Step 12

Evaluate the second derivative.

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Multiply $8$ by $0$ .

$64 \cos (0)$

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

$64 \cdot 1$

Multiply $64$ by $1$ .

$64$

Step 13

$x = 0$ 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 14

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

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

$f (0) = - \cos (8 (0))$

Simplify the result.

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Multiply $8$ by $0$ .

$f (0) = - \cos (0)$

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

$f (0) = - 1 \cdot 1$

Multiply $- 1$ by $1$ .

$f (0) = - 1$

The final answer is $- 1$ .

$y = - 1$

Step 15

Evaluate the second derivative at $x = \frac{π}{8}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$64 \cos (8 (\frac{π}{8}))$

Step 16

Evaluate the second derivative.

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Cancel the common factor of $8$ .

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Cancel the common factor.

$64 \cos (8 \frac{π}{8})$

Rewrite the expression.

$64 \cos (π)$

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.

$64 (- \cos (0))$

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

$64 (- 1 \cdot 1)$

Multiply $64 (- 1 \cdot 1)$ .

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Multiply $- 1$ by $1$ .

$64 \cdot - 1$

Multiply $64$ by $- 1$ .

$- 64$

Step 17

$x = \frac{π}{8}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = \frac{π}{8}$ is a local maximum

Step 18

Find the y-value when $x = \frac{π}{8}$ .

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Replace the variable $x$ with $\frac{π}{8}$ in the expression.

$f (\frac{π}{8}) = - \cos (8 (\frac{π}{8}))$

Simplify the result.

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Cancel the common factor of $8$ .

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Cancel the common factor.

$f (\frac{π}{8}) = - \cos (8 (\frac{π}{8}))$

Rewrite the expression.

$f (\frac{π}{8}) = - \cos (π)$

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.

$f (\frac{π}{8}) = \cos (0)$

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

$f (\frac{π}{8}) = - (- 1 \cdot 1)$

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

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Multiply $- 1$ by $1$ .

$f (\frac{π}{8}) = 1$

Multiply $- 1$ by $- 1$ .

$f (\frac{π}{8}) = 1$

The final answer is $1$ .

$y = 1$

Step 19

These are the local extrema for $f (x) = - \cos (8 x)$ .

$(0, - 1)$ is a local minima

$(\frac{π}{8}, 1)$ is a local maxima

Step 20