First we find the derivative, using the quotient rule:
We're leaving the derivative in this form, with the numerator and denominator factored, because it will make the next pieces of the task easier.
Next we need to find the x-values at which the derivative is undefined but f is defined. Since f'(x) is a rational function, it's undefined wherever the denominator is 0.
(x + 2)2 = 0
we need to have
x + 2 = 0
x = -2,
so f' is undefined at x = -2. However, f is also undefined at x = -2, so x = -2 is not a critical point. Finally, we need to find the roots of f'. Since f'(x) is a rational function, it will be 0 when the numerator is 0 and the denominator isn't 0.In order to have
x(x + 4) = 0
we must have either x = 0 or x = -4. Since neither of these values make the denominator of f' equal zero, x = 0 and x = -4 are critical points. If we look at the graph of f, this makes sense: