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

Jun 1, 2018

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, $f \left(x\right) = \frac{x}{{x}^{2} + 4}$, so $f ' \left(x\right) = \frac{4 - {x}^{2}}{{x}^{2} + 4} ^ 2$ by the quotient rule (and a little combining of terms).

This equals 0 either when the denominator equals $\infty$ (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: $x = \pm 2$, which equate to $f \left(x\right) = \pm \frac{1}{4}$.