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**Glitch** **The question:**

Locate and identify the stationary points for $\displaystyle y = \frac{x}{1+x^2}$

**My attempt:**

Using quotient rule...

$\displaystyle \frac{1 - x^2}{(1 + x^2)^2}$

I find when this equals zero, i.e. $\displaystyle 1 - x^2 = 0$

Therefore, $\displaystyle x = \pm1$

I substitute this into the original eqn to get 1/2 and -1/2. So the points are (1, 1/2) and (-1, -1/2). So far so good.

Now I want to know the nature of these points. So I go to find the second derivative...

$\displaystyle \frac{-2x(1+x^2)^2 - (1 - x^2)4x(1 + x^2)}{(1 + x^2)^4}$

Looks a bit messy!

I substitute 1 into the equation and get a negative number, so I gather that (1, 1/2) is a maximum point. However, when I substitute -1, I get 0.