Intervals of increase/decrease with a exponential function

[endeavor]

find the intervals of increase and decrease of f(x) = e^3x + e^-2x.

f'(x) = 3e^3x - 2e^-2x
I set f'(x) = 0 to find the critical numbers:
3e^3x = 2e^-2x
3 ln e^3x = 2 ln e^-2x
9x = -4x
x = 0, which is obviously wrong, (3e^0 - 2e^0 = 1). I found out that I had to combine the two terms using a common denomintor, and I got the right answer.

But why did my original method fail?

 

[Hootenanny]

Quite simply because;

3e^3x = 2e^-2x
3 ln e^3x = 2 ln e^-2x

$$3\ln|e^{3x}| \neq \ln |3e^{3x}|$$

 

[endeavor]

ahh... i feel so stupid.. i hate when i make simple mistakes and mix formulas up. :uhh:

 

[Hootenanny]

ahh... i feel so stupid.. i hate when i make simple mistakes and mix formulas up. :uhh:

I did a very similar thing in an exam yesterday, I was sat there for fifteen mintues trying to figure out where I had gone wrong, then I realized, I had differentiated instead of integrated , it was a good job I realized before the end of the test!