Question

How do you find the critical points of: `f(x)=(x)(e^(-x^2))`?

Answer 1

See the explanation section, below.

Explanation:

`f(x)=(x)(e^(-x^2))`

`f'(x)=(1)(e^(-x^2)) + (x)(e^(-x^2)(-2x))` (product and chain rules)

`= e^(-x^2)(1-2x^2)`

`f'(x)` is never undefined and is `0` at `x = +-1/sqrt2`.

Both `1/sqrt2` ans `-1/sqrt2` are in the domain of `f` (Domain is `RR`), so both are critical numbers for `f`.

Note

Some treatments have "critical points" being points in 2-space, so we also find `y = f(x)` at these critical numbers. I do not use that terminology, but if your teacher does, then you should find `(1/sqrt2 , 1/(sqrt2root4e))` and `(-1/sqrt2 , -1/(sqrt2root4e))` (or whatever form is expected).