Compute the 2×2 Hessian matrix of second partial derivatives for f(x,y) polynomials.
The Hessian matrix H contains all second partial derivatives of f(x,y). For a 2D function: H = [[∂²f/∂x², ∂²f/∂x∂y], [∂²f/∂y∂x, ∂²f/∂y²]]. At a critical point (where ∇f = 0), det(H) > 0 with fxx > 0 indicates a local minimum, det(H) > 0 with fxx < 0 a local maximum, and det(H) < 0 a saddle point.
Hessian matrix
H = [[fxx, fxy], [fxy, fyy]]
Determinant test
det(H) = fxx * fyy - fxy²
Only when the gradient is zero at that point. The second derivative test uses the Hessian determinant and fxx to classify minima, maxima, and saddle points.
The test is inconclusive — the point may be a degenerate saddle, flat region, or higher-order extremum requiring further analysis.