My teacher has written this problem. Find the extreme values of the function f(x,y) = x^4 + y^4 - 2(x - y)^2 ≤ 0. But what does the "≤ 0" mean? I know literally it means less than 0 but it seems like an unusual function. In addition, he gave us some hints which I don't understand at all. Here it is, from the partial derivatives, f_x = 4x^3 - 4(x-y) = 0 f_y = 4y^3 + 4(x-y) = 0 we get (x,y) = (0,0),(√2,-√2),(-√2,√2) for O,A,B. (I think A is is f_xx(x,y) and B is f_xy(x,y) = 2 , but I am not sure because he didn't say anything) At the origin O, = (0,0), f_xx(0,0)x^2 + z.f_xy(0,0)xy + f_yy(0,0)y^2 = -4x^2 + 8xy - 4y^2 = -4(x^2 - 2xy +y^2) = -4(x-y)^2 ≤ 0 So that's the hint that he gave to the class. Can you explain this because I don't understand anything from it. It's different than how I would usually find extreme values.