aligned & f(x, y)=2(x-y)^2-x^4-y^4 & | (f_ x x f_ y y -f_ x y ^2 ) |_ (0,0) aligned

Step-by-step solution

Given that, f(x, y)=2(x-y)^2-x^4-y^4, we get On differentiating partially w.r.t. x , f_x=4(x-y)-4 x^3 Again differentiating partially, we get aligned & & f_ x x & =4-12 x^2 & & (f_ x x )_ (0,0) & =4-0=4 Similarly & & f_ y y & =4-12 y^2 & & (f_ y y )_ (0,0) & =4-0=4 and & & f_ x y & =-4+0 & & (f_ x y )_ (0,0) & =-4 & & (f_ x x f_ y y -f_ x y ^2 )_ (0,0) & =4(4)-(-4)^2=0 aligned