Dear xstryker,
The problem with the 2nd derivative test is that when its
value is zero, there is no conclusion to be made as to the nature
of the turning point. We consider two examples below.
Suppose we take y = x^3, of which both dy/dx
and (d^2 y) / (d x^2) are 0 (you may verify them). In this
case, we know from the shape of the graph that the
origin is a stationary point of inflexion.
Now take y = x^4. This time, the origin is a minimum
turning point despite having both dy/dx and (d^2 y) / (d
x^2) equal to 0.
The first derivative test is a more complete test that
helps to identify all types of nature of stationary points.
Hope this clarifies.
Cheers,
Wen Shih