Question

How do you find the critical points of the function `f(x) = x / (x^2 + 4)`?

Answer 1

Find those points at which the derivative of `f` is equal to 0

Explanation:

The critical points of a function are those points where its first derivative is 0, i.e. those points where the function reaches a maximum, a minimum, or a point of inflection.

In this case, `f(x)=x/(x^2+4)`, so `f'(x)=(4-x^2)/(x^2+4)^2` by the quotient rule (and a little combining of terms).

This equals 0 either when the denominator equals `oo` (which doesn't happen here for non-infinite `x`) or when the numerator equals 0.

So we want `4-x^2=0`, which tells us the two critical points of the function: `x=+-2`, which equate to `f(x)=+-1/4`.