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

1 Answer
Jun 1, 2018

Answer:

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#.