Determining the type of critical point when the second derivative test fails

[kevinlightman]

Homework Statement

For example with f(x,y) = x²y + xy²

Homework Equations

The Attempt at a Solution

Well I know there is a critical point at (0,0). So I calculated the second derivatives but they are all 0 here so that doesn't help.
I also tried using the Taylor expansion to show that f(x,y)>f(0,0) or not but that didn't get me anywhere.
Then I tried considering the type of critical point on x=0,y=0,y=-x etc. but again I didn't get anywhere.
Any ideas?

 

[Gib Z]

Draw up a table of values to compare values, and most importantly the signs, of the derivatives in the neighborhood of the critical points. It is the same reasoning as is the single variable case eg When we conclude that a critical point was a local maximum if the derivative from the left is positive and the derivative from the right is negative.