Differentiating exponential functions?
Hi Folks. Im slightly embarassed to ask this, since I normally okay with math. but i’m slightly rusty, could do with a little help;
we need to find critical points of f(x), and comment on whether it is a local maximum or minimum.
f(x) = (x)exp(-x^2) [1]
[If f '(x) = 0, and if f''(x) < 0, then it is a maximum, and if f''(x) > 0 it is a minimum (if f''=0, then it is an inflexion point)].
In terms of finding the first derivative, I have got as far as (using product rule):
(-2x^2 +1) exp(-x^2) = 0 [2]
but got stuck with solving for f’(x) = 0. Am i on the right track if I apply log rules (as ln is the inverse of exp) and take it from here?
ln(exp(-x^2)) = ln (-2x^2 + 1)
in which case, it becomes
- x^2 = ln(-2x^2 + 1)?
many thanks
sk
You’re close.
(-2x^2 +1) exp(-x^2) = 0
The thing is, all values of exp are greater than 0. In other words exp(-x^2) > 0 no matter what value of x you use. So you can divide both sides by exp(-x^2), getting
-2x^2 +1 = 0.
You seem to be able to work the rest of this out.
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