Question

What are the critical values, if any, of `f(x)= (2-x)/(x+2)^3`?

Answer 1

A critical point on a function `f(x)` occurs at `x_o` if and only if either `f '(x_o)` is zero or the derivative doesn't exist.

Hence we need to compute the first derivative

`df(x)/dx=d((x-2)/(x+2)^3)/dx=(x-4)/(x+2)^3`

which nullifies at `x_o=4`

Using the second derivative test for local extrema we need to compute the second derivative for `f(x)` hence we have that

`d^2f(x)/(d^2x)=-6*(x-6)/(x+2)^5`

Because `f''(4)>0` hence the function has a minimum at `x=4`
which is `f(4)=-1/108`

It does not have a maximum as

`lim_(x->-2) f(x)=oo`