Question

f(x)= x^2 / (x-2)^2
find the intervals of concavity and the inflections points

Answer 1

I will be happy to critique your work.

Answer 2

If the function is twice differentiable within its domain, inflection points occur at values of c where f"(c)=0.
Note that not all points where f"(c)=0 are inflection points.

You need to check the concavity <i>between</i> points where f"(x)=0. It is concave up if f"(x)>0 and concave down if f"(x)<0.

X=c is an inflection point if all of the following conditions are satisfied:
1. f(x) is twice differentiable at c.
2. f"(c)=0
3. f"(c+) has a different sign than f"(c-), i.e. concavity changes.

Now it's time for you to sharpen your pencil and show us some work.