Question

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

Answer 1

Critical point:
A critical point at the interior of the domain of a function is the point where the derivative is <i>zero</i> or <i>undefined</i>.


f(θ) = 2cos(θ) + (sin(θ))^2
does not have any undefined points for θ∈ℝ.
So we only need to find the values of θ which make f'(θ)=0.
Differentiate f(θ) and equate to zero. Solve for <i>all</i> roots for the equation and these are the critical points.

The following graph might help you check you answer:
http://img256.imageshack.us/img256/11/1291497419.png

Answer 2

solve on the interval of 0,2π, cos θ / 3 =-1