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

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.

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.
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 ($$t=0$$): $$f(-1)=1;\\ f(3)=-1 \ \text{(min)};\\ f(0)=2 \ \text{(max)}.$$