- None
- One
- Two
- Three
- Four

Solution :

$f$ 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?

**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.

- None
- One
- Two
- Three
- Four

Solution :

$f$ 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.

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This webpage is LaTeX enabled. To type in-line formulae, type your stuff between two '$'. To type centred formulae, type '\[' at the beginning of your formula and '\]' at the end.

How could you post a latex question and solution on your page? I liked the interface. Is there a template for this.

Regards,

Ozhan

The code for making spoilers in your posts can be found here. For the LaTeX, I use mathjax (by googling MathJax on Blogger, I think you can get sufficient enough detail on how to install it on your own blog).

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