Answer to Question #263128 in Calculus for guy

Question #263128

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

𝑓(πœƒ)=18cos(πœƒ)+9sin^2(πœƒ), βˆ’πœ‹β‰€πœƒβ‰€πœ‹



1
Expert's answer
2021-11-09T16:51:26-0500

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"


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