To find: absolute maximum value.
We differentiate to find any critical numbers:
f'x=1-ex. We see that f'x>0 whenever 1-ex>0, that is, 1>ex, that is,x<0. Similarly, f'x<0 whenever 1<ex, that is,x>0. By the First Derivative Test for absolute extreme values, f has an absolute maximum value whenever x=0 and the value is f0=0-e0=-1.