Question

The function g(θ ) = cos2 (θ ) fxx (x0 , y0 ) + 2 cos(θ)sin(θ ) fxy (x0 , y0 ) + sin2 (ϑ ) fyy (x0 , y0 ) represents the Gauss curvature of the surface f (x, y) at the critical point (x0 , y0 ) in the direction (cos(θ ),sin(θ ),0), where 0 ≤θ ≤ 2π. Show that the product of the maximum and minimum values of g(θ ) is fxx (x0 , y0 ) fyy (x0 , y0 ) − fxy 2 (x0 , y0)