Finding critical number for derivative of exponential?
Ok, so I have a function that was originally f (x) = e^2x - 3e^x + 2....the problem asks me to find X intercepts, Y intercepts, relative extrema, inflection points, concavity and any horizontal asymptotes...and if you're wondering why all that, it's so I can finally graph it.
So I factored it and determined that it's (e^x - 1) * (e^x - 2), so e^x = 1 and e^x = 2, which means x = ln1 and x = ln2, so there's X-intercepts at (0,0) and (0.693,0). So I got that ok.
Next for the Y-int, I set x equal to 0 and solved for it, and determined e^2(0) - 3e^(0) + 2 = 0 so Y-intercept is also at (0,0).
Now for the relative extrema, this is where I got stuck. I determined the derivative to be f ' (x) = 2e^2x - 3e^x.....and this is where I'm stuck. What would be the best way to factor this out so I can get the critical numbers of the first derivative? If I factor out a e^x, then I have e^x ( 2e^x - 3) which doesn't necessarily help me since if you set the stuff inside parentheses to x=0 then you have ln3/2 but the e^x on the outside, well even if e^(0), it still equals 1...so yeah, I'm probably overlooking something simple, but what are the critical numbers for the first derivative? And more importantly, how did you find them? Thanks guys!