Find the critical numbers of the function on the interval 0 ≤ θ < 2π. f(θ) = 2cos(θ) + sin2(θ)

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

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.

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Thanks!