Question

Find the critical numbers of the function on the interval 0 ≤ θ < 2π. f(θ) = 2cos(θ) + (sin(θ))^2??

Answer 1

To find critical numbers you want to take the derivative and set it equal to 0.
f(θ) = 2cos(θ) + (sin(θ))^2
f '(θ) = -2sin(θ) + 2sin(θ)cos(θ)

Answer 2

by critical number you mean the angle?

Answer 3

yea it is just cos and sin is what gets me. they confuse me.

Answer 4

critical numbers are where f'(x) is either 0 or undefined, so first we need the derivative using the chain rule:
f'(x)=-2sinx+2sinx(cosx)=0
cosx=1
x=2(pi)n where n=0,1,2,...

Answer 5

0 = -2sin(θ) + 2sin(θ)cos(θ)
0 = 2sin(θ)(-1 + cos(θ))
So you either want
cos(θ) = 1
or
sin(θ)=0

Answer 6

cos(θ) = 1 at θ = π/2
or
sin(θ)=0 at θ = 0, π
The critical values should be 0, π/2, and π