Maximum and Minimum Values using Trigonometric Functions: $f(t)=t-2\cos(t)$ for $t\in[-\pi, \pi]$

Question

RESOLVED

How would I go about finding the maximum and minimum values for an expression I cannot factor to find zeros?

I have some function:

$$f(t)=t-2\cos(t)$$

With the interval $[-\pi, \pi]$.

I would like to know how to find the absolute maximum and absolute minimum values specifically.

Answer 1

Based on the comments and your answer above, I assume you understand the process of finding absolute extreme and are just having trouble with finding the critical values. We found the derivative to be $f(t)=1+2sin(t)$, and by definition of critical numbers we want to see when this is equal to zero. Rearranging, we have $sin(t)=-1/2$, and now we have to use our knowledge of the unit circle. We are really trying to think backwards, and ask ourselves the sin of what, or in other words $t=arcsin(-1/2)$. This gives us $t=-5\pi/6$, and $t=\pi/6$. From now follow tips given by Mark to find the extreme points.

Answer 2

HINT:

Note that

$$f'(t)=1+2\sin(t)=0$$

when $t=-\pi/6$ and $t=-5\pi/6$.

At which of these values of $t$ is $f(t)$ a local maximum?

What are the values of $f(t)$ at the end points $-\pi$ and $\pi$?