Absolute maxima and absolute minima

Question

Consider the function f(x)=2e^-x^(2)). Find the absolute maxima and absolute minima on the interval [-2,1]

This question has more to it, in which I also had to find the relative maxima and relative minima in which I got (0,2) as the relative maxima, there's no relative minima. I'm a bit confused about getting absolute maxima and absolute minima because for the absolute maxima I get (0,2) and for the absolute minima I believe its (-2, 0.04), but when I go on mathway to confirm my answer it says there is no absolute minima, which I guess would make sense because there's only a relative maxima, but for all these type of questions I always test the endpoints and the critical number, so basically what I don't get is why do we ignore absolute minima, or can I also put absolute minima as well.

Answer 1

Mathway is goofed up. Every function (that is defined and continuous on an interval) has an absolute maximum and an absolute minimum on that interval; this is called the extreme values theorem.

Sometimes the absolute max or min is at the endpoints of the interval and sometimes in between. Those that are in between are found using the derivative (either looking at where the derivative DNE meaning the absolute min/max DNE or setting it equal to zero and finding the traditional max/min).

Looks like you are doing things just right. Just make sure you look at places on the interval where the function has a vertical asymptote (derivative DNE). You may need to answer some questions with the exact value, in this case the absolute minimum (occurring at x = -2) is 2e^-4 (an irrational number).