Question

List the critical numbers of the following function in increasing order. Enter N in any blank that you don't need to use.

𝑓(𝜃)=18cos(𝜃)+9sin^2(𝜃), −𝜋≤𝜃≤𝜋

Answer 1

Let us list the critical points of the following function in increasing order:

$f(\theta)=18\cos(\theta)+9\sin^2(\theta), \ -\pi\le \theta\le \pi.$

Since $f'(\theta)=-18\sin(\theta)+18\sin(\theta)\cos(\theta)=18\sin(\theta)(-1+\cos(\theta)),$ we conclude that $f'(\theta)=0$ implies $\sin(\theta)=0$ or $\cos(\theta)=1.$ It follows that $\theta=\pi n,n\in\Z$ or $\theta=2\pi m,m\in\Z.$

Taking into account that $-\pi\le \theta\le \pi,$ we conclude that the critical points are $-\pi,\ 0,\ \pi.$

Answer: $-\pi,\ 0,\ \pi$