MATH SOLVE

2 months ago

Q:
# Find the critical numbers of the function. (Enter your answers as a comma-separated list. Use n to denote any arbitrary integer values. If an answer does not exist, enter DNE.) f(θ) = 10 cos(θ) + 5 sin2(θ)

Accepted Solution

A:

If[tex]f(\theta)=10\cos\theta+5\sin^2\theta[/tex]then the derivative is[tex]f'(\theta)=-10\sin\theta+10\sin\theta\cos\theta[/tex]Critical points occur where [tex]f'(\theta)=0[/tex]. This happens for[tex]-10\sin\theta+10\sin\theta\cos\theta=0[/tex][tex]-10\sin\theta(1-\cos\theta)=0[/tex][tex]\implies-10\sin\theta=0\text{ or }1-\cos\theta=0[/tex]In the first case, we find[tex]-10\sin\theta=0\implies\sin\theta=0\implies\theta=n\pi[/tex]In the second,[tex]1-\cos\theta=0\implies\cos\theta=1\implies\theta=2n\pi[/tex]So all the critical points occur at multiples of [tex]\pi[/tex], or [tex]n\pi[/tex]. (This includes all the even multiples of [tex]\pi[/tex].)