1. The problem statement, all variables and given/known data 1. What happens to D = fxxfyy - (fxy)2 at (0,0) for f(x,y) = 9x4 - 6x2y2 + y4? Classify the critical point at (0,0). 2. How about if f(x,y) = (y - x2)(y - x4) ? 2. Relevant equations See above ^. 3. The attempt at a solution 1. Okay, so after taking all the partial derivatives and computing D, we get that D = 0 at (0,0) so that the second partial test fails. So we have to consider some other method to see what happens at the origin. Suppose we set y=0, so that f(x,0) = 9x4. Then fx(x,0) = 36x3. Now if x < 0, fx < 0 and if x > 0, fx > 0. So it is clear that (0,0) acts like an inflection point along the x-axis and we can conclude that (0,0) is a minimum? I'm not sure about my logic here since it's my first time trying one of these. Anyone have any pointers? 2. Same process as before, I take all my derivatives and compute D to get D = 0 so that at (0,0) my test fails. Again I'll suppose that y = 0, so that f(x,0) = x6 and fx(x,0) = 6x5. Now if x < 0, fx < 0 and if x > 0, fx > 0. So it is clear that (0,0) acts like an inflection point along the x-axis once again and we can conclude that (0,0) is a minimum? I'm hoping my reasoning is correct.