## fr0568 #63

Forum for the GRE subject test in mathematics.
rhnsrbh
Posts: 5
Joined: Sat Mar 27, 2010 5:03 am

### fr0568 #63

63 If f is the function defined by f(x)= x*exp(-x^2-x^(-2) if x!=0 and =0 if x=0, at how many values of x does the graph of f have a horizontal tangent line?

I've got f'(0) = 0 from the definition of a derivative at x = 0. I'm having trouble finding the other two roots of f'(x). I have f'(x) = Exp(-x^2-x^-2)*(1+2x^2-2x^-2), so if f'(x) = 0, (1+2x^2-2x^-2) = 0, but I can't find the roots to this. In particular, -1 and 1 don't seem to work. I don't think I'm calculating the derivative incorrectly. Any ideas?

enork
Posts: 33
Joined: Fri Sep 18, 2009 3:16 am

### Re: fr0568 #63

To find the roots of $1+2x^2-2x^{-2}=0$ multiply through by $x^2$ and substitute $y=x^2$. Then you'll have a quadratic $y+2y^2-2=0$ that you can use the quadratic formula on. However, I'm not sure the signs are right on the derivative.

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