unusual critical point

consider the following function:

f(x,y) = ( y² - x³ ) ( y² - 2x³ )

1. Establish that (0, 0) is a critical point for f. Does f have any other critical points?

2. Show that the second-order partial derivative test fails to establish whether f has a local maximum, local minimum, or a saddle point at this critical point.

3. Examine the behavior of f(x, y) along every line in the XY plane passing through the origin. Does anything special happen there?

4. What does f have at the origin, a maximum, a minimum, or a saddle point? Hint: Where is f positive, where is f negative, and where is f equal to 0?

5. Explain intuitively what is going on. Explain what is strange here and how to reconcile it. Graph f in three dimensions, and explain in words what the graph looks like.