Infimum and supremum, min and max?

Note that f(x) = (e^x - x)/x is a continuous function on [0, 3], which is a closed interval. Hence, f attains its extreme values, and the maximum/supremum and minimum/infimum coincide.

Critical points:
f'(x) = (e^x * e^x - (e^x - x) * e^x)/(e^x)^2 = xe^x/e^(2x) = xe^(-x).
This equals 0 only when x = 0 (an endpoint).

Thus, the extreme values occur at the endpoints:
f(0) = 1 <---Maximum
f(3) = (e^3 - 3)/e^3 <---Minimum

I hope this helps!