find the critical numbers of the function. (enter your answers as a comma - separated list. use ( n ) to…

Explanation:

Step1: Find the derivative of ( f(\theta) )

Use the chain - rule. The derivative of ( y = \cos(u) ) is ( y^\prime=-\sin(u)\cdot u^\prime ) and the derivative of ( y = u^{2} ) is ( y^\prime = 2u\cdot u^\prime ). For ( f(\theta)=16\cos(\theta)+8\sin^{2}(\theta) ), we have ( f^\prime(\theta)=-16\sin(\theta)+16\sin(\theta)\cos(\theta)). Factor out ( - 16\sin(\theta)): ( f^\prime(\theta)=-16\sin(\theta)(1 - \cos(\theta))).

Step2: Set ( f^\prime(\theta)=0 )

Set ( -16\sin(\theta)(1 - \cos(\theta)) = 0 ). Since ( -16\neq0 ), we consider two cases:

Case1: ( \sin(\theta)=0 )

The general solution of ( \sin(\theta)=0 ) is ( \theta = n\pi), where ( n\in\mathbb{Z}).

Case2: ( 1-\cos(\theta)=0 )

If ( 1-\cos(\theta)=0 ), then ( \cos(\theta)=1 ). The general solution of ( \cos(\theta)=1 ) is ( \theta = 2n\pi), where ( n\in\mathbb{Z}). The solutions of ( \theta = 2n\pi) are included in the solutions of ( \theta=n\pi) (when ( n = 2k,k\in\mathbb{Z})).

Answer:

(n\pi)