For the function, use the second derivative test (if possible) to determine if each critical point is a minimum, maximum, or neither. If the second derivative test can't be used, say so.
The first derivative is
The only critical point is x = 1, since x = 0 isn't in the domain of f and ex is never 0.The second derivative is
We could simplify this, but it's probably not worth it: all we need to do is plug in 1 for x and see what we find, so simplifying first would be a waste of time.
The function f is concave up at x = 1, and so has a minimum there.