need explanation thanks!

f(θ) = 2cos(θ) + sin2(θ)
f '(θ) = 2sin(θ) + 2cos(2θ) = 0
=> sin(θ) = cos(2θ)
=> sin(θ) = 1  2sin^2(θ)
=> 2sin^2(θ) + sin(θ)  1 = 0
=> (2sin(θ)  1)(sin (θ) + 1) = 0
sin(θ) = 1 and 1/2
θ = Π/6, 5Π/6 and 3Π/2
f '(θ) = 2sin(θ) + 2cos(2θ) = 0
=> sin(θ) = cos(2θ)
=> sin(θ) = 1  2sin^2(θ)
=> 2sin^2(θ) + sin(θ)  1 = 0
=> (2sin(θ)  1)(sin (θ) + 1) = 0
sin(θ) = 1 and 1/2
θ = Π/6, 5Π/6 and 3Π/2

I agree with mohanrao with the answer/method. However, keep in mind that critical numbers are defined as places where f '(x) =0 OR undefined. For example, if you had a rational function with an x on the denominator as f '(x), you would have to find where the numerator was equal to 0 AND state that x=0 is also a critical number (because anything divided by 0 is undefined) While this problem doesn't have a situation where f '(x) is undefined, it is good to keep in mind that  f '(x) = undefined  is a definition of a critical number.

To make learning math a bit easier, Dr. Pan (TucsonMathDoc) has recorded a YouTube video to help visually answer your question.
Please comment on YouTube or Y!A and let her know if it helped.
Thanks!
Please comment on YouTube or Y!A and let her know if it helped.
Thanks!