Maximum and minimum values

Question

For the following functions, find all critical points exactly.

1. f(x)= e^-1 - e^-3x for x>0

Answer 1

f(x) = e^-x - e^-3x

f'(x) = -e^-x + 3e^-3x = 0

-e^-x + 3e^-3x = 0

e^-x(-1 + 3e^-2x) = 0

e^-x > 0 for all x. So, -1 + 3e^-2x = 0.

e^-2x = 1/3 So, -2x = ln(1/3) = ln1 - ln3 = -ln3.

Therefore, x = (1/2)ln3 = ln√3

Answer 2

Only /=one critical point. (1/3) is a zero.