Does this function have 2 critical numbers? e^x + e^2x?

f(x) = e^x+e^(2x)
f'(x) = e^x+2e^(2x) = e^x+2(e^x)^2 = 0
e^x(2e^x+1) = 0
e^x = -1/2 and 0
No solution
No critical numbers

No. It has positive slope everywhere.

y = e^x + e^(2x)
y' = e^x + 2e^(2x)
e^x + 2e^(2x) = 0
e^x = -2e^(2x)
(-2e^(2x)) / (e^x) = 1
-2e^x = 1
e^x = -1/2
x = 2 + i*pi*n, for any odd number n
There are infinity critical numbers, but none of them are real.

e^x - 2e^(2x) = 0---->e^x = 1/2 --->x=-ln(2)