Find the absolute minimum value of the function gx ex5x x

Find the absolute minimum value of the function g(x) = e^x/(5x), x > 0.

The critical values would be where the derivative is 0, or at the endpoint, x = 0.

Thus, taking the derivative, using division rule,

g\'(x) = [5x (e^x) - (e^x)(5)] / (5x)^2

Setting this to 0,

[5x (e^x) - (e^x)(5)] / (5x)^2 = 0

[5x (e^x) - (e^x)(5)] = 0

5x - 5 = 0

x = 1

Evaluating at x = 0 and x = 1:

g(0) = undefined
g(1) = e/5

Thus, the absolute minimum value of g(x) is = e/5. [ANSWER]