Math GRE - #34

$f$ is the function defined by: $$f(x)=\begin{cases} xe^{-x^{2}-x^{-2}} & \text{if } x\neq0\\ 0 & \text{otherwise.}\end{cases}$$ At how many values of $x$ does $f$ have a horizontal tangent line?

1. None
2. One
3. Two
4. Three
5. Four

Solution :

Choice 4 is the answer.

$f$ has a horizontal tangent line only when the derivative of $x$ is 0. The derivative of $f$ is:
$$f^\prime(x) = e^{-x^2-x^{-2}}+(-2x+2x^{-3})xe^{-x^2-x^{-2}}=(1-2x^2+2x^{-2})e^{-x^2-x^{-2}}=0.$$ This is only zero in two cases: in the limit $\displaystyle\lim_{x\rightarrow 0} f^\prime(x)$ and when $1-2x^2+2x^{-2}=0$.

The quadratic-like equation above can be solved by solving $x^2-2x^4+2=0.$ This equation has two non-zero solutions. These solutions, along with the with the solution obtained in the limit above, gives us three horizontal tangent lines in total.