# Hessian Matrix. Second partial derivative test

I have a question about the hessian matrix / determinant of the hessian matrix. For the Hessian matrix, one is able to determine the local curvature around a point. And with the Second partial derivative test one can test, using the determinant of the hessian, if that point is located on an extrema, or saddle point.

My question is Wikipedia states:

the Hessian approximates the function at a critical point with a second degree polynomial.

By 'hessian' does this mean that the hessian matrix approximates the function or the determinant? (i am confused as determinant of hessian and hessian matrix are sometimes used intertwined)

thank you in advance

• The Hessian MATRIX is used in a second order Taylor approximation. – Nir Regev Jan 6 '16 at 10:49

## 1 Answer

The function is fully approximated if one uses all the derivatives (see Taylor expansion). With using the Hessian only, we can only make a second degree approximation (because it is second derivative matrix), which is geometrically the same as using the second order polynomial.