**f(x)= x^2/(x-2)^2**

Find:

a) Vertical and horizontal asymptotes.

b) Intervals of increase or decrease.

c) local maximum and minimum values.

d) intervals of concavity and inflection points

e) Sketch the graph of f(x)

a) Find the values that \(\displaystyle x\) and \(\displaystyle f(x)\) can not take.

b) Derivative \(\displaystyle > 0\) means increasing.

Derivative \(\displaystyle < 0\) means decreasing.

c) Derivative \(\displaystyle =0\) means a stationary point.

If the second derivative \(\displaystyle >0\) at that point, you have a minimum.

If the second derivative \(\displaystyle <0\) at that point, you have a maximum.

If the second derivative \(\displaystyle =0\) the test is inconclusive. Check the derivative at values near that point.

d) The function is concave when the second derivative \(\displaystyle < 0\).

The function is convex when the second derivative \(\displaystyle > 0\).

The function has inflection points when it goes from convex to concave or vice versa. I.e. where the second derivative \(\displaystyle = 0\).

e) Use the information found in a) - d).