Intervals on which function is decreasing and increasing

Hey, could someone please guide me as to what to do next in this problem. Thanks!

f(x) = xe^{-(x^2)/2}

Determine the intervals where f is increasing and where f is decreasing.

I've worked out the derivative, but am unsure how to proceed.

Update: I've worked out that the derivative is positive for -1 < x < 1 and negative for x > 1 and x < -1 ... but my working seems quite dodgy, just a bit of guidance would be great. Thanks.

Re: Intervals on which function is decreasing and increasing

You are correct, but how did you determine this...what did you get for the derivative?

Re: Intervals on which function is decreasing and increasing

I got f'(x) = e^{(-x^2)/2}(1+x)(1-x)

The I set f'(x) > 0 and solved, and did the same for f'(x) < 0

Re: Intervals on which function is decreasing and increasing

I see nothing "dodgy" about that. I would probably write:

$\displaystyle f'(x)=e^{-\frac{x^2}{2}}(1-x^2)$

observe that $\displaystyle e^{-\frac{x^2}{2}}>0$ for all real $\displaystyle x$, and then observe:

$\displaystyle 1-x^2$

is positive on $\displaystyle (-1,1)$ and negative on $\displaystyle (-\infty,-1)\,\cup\,(1.\infty)$.

What you did seems find to me.