If 𝑓(𝑥) = cos2 𝜃 − 2 sin 𝜃 , 0 ≤ 𝜃 ≤ 2𝜋, find where 𝑓 is increasing, decreasing, intervals of concavity and inflection point(s).

Answer 1

df/dθ = -2sin2θ - 2cosθ

Equate to 0 to find critical points: sin2θ = -cosθ or 2sinθcosθ = -cosθ

This occurs when cosθ = 0 at θ = pi/2 and 3pi/2

It also occurs at sinθ = -1/2 at θ = 7pi/6 and 11pi/6

You can check second derivative at those points to determine concavity and you can test regions for sign (there are 5 regions)

df²/dθ² = -4cos2θ + 2sinθ ( less than zero, concave down and greater than 0 concave up.)