Calculator Input
Enter second partial derivatives at a critical point. Then classify the point using the Hessian determinant test.
Example data table
| Example | Point | f_xx | f_yy | f_xy | D = f_xx f_yy - f_xy² | Classification |
|---|---|---|---|---|---|---|
| Paraboloid-type curvature | (1, -1) | 6 | 8 | 2 | 44 | Local Minimum |
| Inverted dome | (0, 0) | -5 | -4 | 1 | 19 | Local Maximum |
| Mixed curvature surface | (0, 0) | 2 | -3 | 1 | -7 | Saddle Point |
| Degenerate case | (2, 1) | 1 | 1 | 1 | 0 | Inconclusive |
Formula used
1) Hessian determinant for two variables
D = f_xx(x₀, y₀) × f_yy(x₀, y₀) - [f_xy(x₀, y₀)]²
2) Classification rules
- If D > 0 and f_xx > 0, the point is a local minimum.
- If D > 0 and f_xx < 0, the point is a local maximum.
- If D < 0, the point is a saddle point.
- If D = 0, the test is inconclusive.
3) Hessian matrix
H = [ [ f_xx, f_xy ], [ f_xy, f_yy ] ]
How to use this calculator
- Find a critical point where first partial derivatives are zero.
- Compute the second partial derivatives at that point.
- Enter x₀, y₀, f_xx, f_yy, and f_xy.
- Choose a tolerance for near-zero determinant checks.
- Set the graph range for the local surface view.
- Click Evaluate Critical Point.
- Read the classification, determinant, eigenvalues, and curvature note.
- Download CSV or PDF if you need a saved result.
Frequently asked questions
1) What does this calculator classify?
It classifies a two-variable critical point as a local minimum, local maximum, saddle point, or inconclusive case using second partial derivatives and the Hessian determinant.
2) Do I need first derivatives first?
Yes. The second derivative test only applies after confirming the point is critical, which means the first partial derivatives are zero at that location.
3) Why does the result say inconclusive?
That happens when the determinant is zero, or very close to zero under your tolerance. In those cases, second-order information alone does not fully classify the point.
4) Why is f_xx used with D > 0?
When D is positive, the Hessian is either positive definite or negative definite. The sign of f_xx distinguishes whether the point is a minimum or maximum.
5) What does the Plotly graph show?
It shows the quadratic approximation near the critical point. This local model helps you see upward curvature, downward curvature, or the mixed shape of a saddle point.
6) Can I use it for three variables?
This version is designed for two variables. Higher-dimensional tests use larger Hessian matrices and different definiteness checks.
7) What do the Hessian eigenvalues mean?
They describe curvature along principal directions. Two positive eigenvalues indicate local upward curvature, two negative eigenvalues indicate local downward curvature, and mixed signs indicate a saddle point.
8) Why add a tolerance value?
Real calculations often contain rounding noise. A tolerance helps treat extremely small determinant values as nearly zero, which avoids misleading classifications from tiny numerical errors.