Find the local maximum and minimum values and saddle points of the function :

f(x,y) = x^3 - 6xy + 8y^3

I found the critical points (0,0) and (1,1/2) by setting the partial derivatives equal to zero and solving for x and y.

Then, I tried the second derivative test:

D = fxx(a, b) * fyy (a, b) - [fxy(a, b)]^2

But found that D(0,0) = 0, meaning that the second derivative test fails for the point (0,0). What do I do when the second derivative test fails? Note that I am not allowed a graphing calculator.

Thank you,

David