Find the coordinates for the absolute maximum and minimum values of the function on the given interval

Question

I am trying to learn how to find the coordinates for the absolute maximum and minimum values of the function on the given interval.

$$f(t) = 2-|t|, -1 ≤ t ≤ 3$$

The answer in my textbook says the minimum is $(3, -1)$ and the maximum is $(0, 2)$.

I keep getting different results, and I don't understand what I'm doing wrong. If someone could provide me with the general concept of how to solve these types of questions, I would really appreciate it.

My solution:

$f(-1) = 2-|-1|$

$f(-1) = 1$

$f(3) = 2 - |3|$

$f(3) = -1$

I thought doing that would give me the $y$ values for the absolute max and min, but clearly, it differs from the answer provided in the textbook.

Additionally, I don't really understand how to get the corresponding $x$ values for the max and min. I tried taking the derivative of the original function which left me with $f'(t) = -t/|t|$ but I didn't know where to go from there.

Answer 1

Hint: Split up $(-1,3)$ into two intervals: $(-1,0)$ and $(0,3)$. On the former interval, since $t<0$, $|t|=-t$ and thus $f(t) = 2 + t$ on that interval. Similarly, you can show $f(t) = 2 - t$ on the second interval.

Then consider the derivative of each function on each interval, and particularly what that derivative means in the scope of the function overall - derivative tests aren't going to help a whole lot here, so think moreso about what the derivative means in a qualitative sense and how it describes the behavior of the function.

Looking at a graph might prove useful for this qualitative analysis in particular so here's one I quickly hashed up on Desmos:

Answer 2

This is a constrained optimization problem with non-linear objective funciton. According to the Extreme Value Theorem, you must check the border points ($t=-1$, $t=3$) and the critical points in the constrained interval ($t=0$): $$f(-1)=1;\\ f(3)=-1 \ \text{(min)};\\ f(0)=2 \ \text{(max)}.$$