Signs of Products, and the Trivial Inequality (x² ≥ 0)

Let $a$ be a nonzero real constant. Given that $x$ is a real number, find the maximum value of: $$\frac{|x|}{x^2+a^2}$$ as well as the value of $x$ at which this maximum occurs.

When you are done, check the answer to see if you are correct!