For the function, find all critical points or determine that no such points exist.

Answer

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.

To have

(*x* + 2)^{2} = 0

we need to have

*x* + 2 = 0

or

*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: