Question

What are the critical points of f(x)= e^x / 3+e^x?

Answer 1

You have posted enough questions here to know that we need brackets to avoid confusion with these kind of fractions.

I will assume you meant
f(x) = e^x/(3+e^x)

By the quotient rule :
f'(x) = [(3+e^x)(e^x) - e^x(e^x)]/(3+e^x)^2
= 3e^x/(3+e^x)^2

set that equal to zero to find max/min points
so 3e^x = 0
no solution
Your function has no max or min points.

take second derivative by using the quotient rule again, set that equal to zero to find any points of inflection.

Some texts will include the x and y intercepts when finding "critical" points, so sub in x=0 to find the y-intercepts
f(0) = e^0/(3+e^0) = 1/(3+1) = 1/4
for the x-intercept, let f(x) = 0
0 = e^x/(3+e^x)
then 0 = e^x
which yields no solution.