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Math  /  Calculus

QuestionList the critical numbers of the following function in increasing order. Enter N in any blank that you don't need to use. f(θ)=4cos⁡(θ)+2sin⁡2(θ),−π≤θ≤πf(\theta)=4 \cos (\theta)+2 \sin ^{2}(\theta),-\pi \leq \theta \leq \pi □\square

Studdy Solution

STEP 1

1. We are given the function f(θ)=4cos⁡(θ)+2sin⁡2(θ) f(\theta) = 4 \cos(\theta) + 2 \sin^2(\theta) .
2. We need to find the critical numbers of this function within the interval −π≤θ≤π-\pi \leq \theta \leq \pi.
3. Critical numbers occur where the derivative is zero or undefined.

STEP 2

1. Find the derivative of f(θ) f(\theta) .
2. Set the derivative equal to zero and solve for θ \theta .
3. Determine if the derivative is undefined for any θ \theta in the given interval.
4. List the critical numbers in increasing order.

STEP 3

Find the derivative of f(θ) f(\theta) .
f(θ)=4cos⁡(θ)+2sin⁡2(θ) f(\theta) = 4 \cos(\theta) + 2 \sin^2(\theta)
Use the chain rule and power rule to differentiate:
f′(θ)=ddθ[4cos⁡(θ)]+ddθ[2sin⁡2(θ)] f'(\theta) = \frac{d}{d\theta}[4 \cos(\theta)] + \frac{d}{d\theta}[2 \sin^2(\theta)]
f′(θ)=−4sin⁡(θ)+4sin⁡(θ)cos⁡(θ) f'(\theta) = -4 \sin(\theta) + 4 \sin(\theta) \cos(\theta)

STEP 4

Set the derivative equal to zero:
−4sin⁡(θ)+4sin⁡(θ)cos⁡(θ)=0 -4 \sin(\theta) + 4 \sin(\theta) \cos(\theta) = 0
Factor out 4sin⁡(θ) 4 \sin(\theta) :
4sin⁡(θ)(−1+cos⁡(θ))=0 4 \sin(\theta)(-1 + \cos(\theta)) = 0
This gives us two equations to solve:
1. sin⁡(θ)=0 \sin(\theta) = 0
2. −1+cos⁡(θ)=0 -1 + \cos(\theta) = 0

STEP 5

Solve sin⁡(θ)=0 \sin(\theta) = 0 :
θ=nπ \theta = n\pi
Within the interval −π≤θ≤π-\pi \leq \theta \leq \pi, the solutions are:
θ=−π,0,π \theta = -\pi, 0, \pi

STEP 6

Solve −1+cos⁡(θ)=0 -1 + \cos(\theta) = 0 :
cos⁡(θ)=1 \cos(\theta) = 1
Within the interval −π≤θ≤π-\pi \leq \theta \leq \pi, the solution is:
θ=0 \theta = 0

STEP 7

Determine if the derivative is undefined for any θ \theta in the given interval. The derivative is defined for all θ \theta in the interval because it is composed of sine and cosine functions, which are defined everywhere.

STEP 8

List the critical numbers in increasing order:
θ=−π,0,π \theta = -\pi, 0, \pi
The critical numbers of the function f(θ) f(\theta) are:
−π,0,π \boxed{-\pi, 0, \pi}

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