Hessian Matrix Calculator
Compute the 2×2 Hessian matrix of second partial derivatives for f(x,y) polynomials.
Inputs
Enter function using x and y as variables.
∂²f/∂x²
6.000005
∂²f/∂x∂y
2.000000
∂²f/∂y²
-10.000003
det(H)
-64.000066
Step by step
Values used
f(x, y) = x^3 + x*y^2 - 2*y^3; x value = 1; y value = 1
Hessian matrix
H = [[fxx, fxy], [fxy, fyy]]
Determinant test
det(H) = fxx * fyy - fxy²
∂²f/∂x²
= 6.000005
∂²f/∂x∂y
= 2.000000
∂²f/∂y²
= -10.000003
det(H)
= -64.000066
How it works
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.
Formulas
Hessian matrix
H = [[fxx, fxy], [fxy, fyy]]
- fxx
- ∂²f/∂x²
- fxy
- ∂²f/∂x∂y
- fyy
- ∂²f/∂y²
Determinant test
det(H) = fxx * fyy - fxy²
- det(H)
- Hessian determinant
Frequently Asked Questions
When can I classify a critical point?
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.
What if the determinant is zero?
The test is inconclusive — the point may be a degenerate saddle, flat region, or higher-order extremum requiring further analysis.