Question

Consider the function `y = e^x` defined on `[-1, 2]`. What are the maximum and minimum of the function on this interval?

Answer 1

This function has no absolute maximum or minimum, but would have a local minimum at `y = 1/e` and a local maximum at `y = e^2` on the interval `[-1, 2]`

Explanation:

We don't need calculus to explain this. The graph of any exponential function will have a domain of all the real numbers. The graph of `e^x` will have a range of `y>0`, however the graph will never touch `y=0`, therefore `y = 0` cannot be considered a minimum.

In calculus speak, we would say `lim_(x-> -oo) e^x = 0`, which means that the function approaches `y = 0` as `x` approaches negative infinity.

So the local max/min will be the two end points of our closed interval. Hence there willl be a local maximum at `x = 2` and local minimum at `x = -1`

Hopefully this helps!