# Evaluate list of determinant of Hessian at list of points

If f:= x^2y + z^3 + xz; and if {{1,2,9},{2,4,6},{9,5,9},{3,3,1},{6,5,3}} is a list of points. I need to evaluate the hessian at each point and then evaluate the determinant at each hessian at the same time.

I have tried the following code to evaluate hessian at $5$ points

f[x_, y_, z_] :=  f:= x^2y + z^3 + xz;

hessianF[x_,y_,z_] = hessian[x, y, z][f[x, y, z]]//FullSimplify

hessianF @@@ {{1,2,9},{2,4,6},{9,5,9},{3,3,1},{6,5,3}}


Thanks for any help

• D[f[x, y, z], {{x, y, z}, 2}] gives the Hessian. – J. M.'s ennui Oct 20 '15 at 9:58
• Did you forget to include the code for hessian? – Daniel Lichtblau Oct 20 '15 at 14:51

f = x^2 y + z^3 + x z;
hessian[x_, y_, z_] = D[f, {{x, y, z}, 2}]
{{2 y, 2 x, 1}, {2 x, 0, 0}, {1, 0, 6 z}}

m = {{1, 2, 9}, {2, 4, 6}, {9, 5, 9}, {3, 3, 1}, {6, 5, 3}};

h = hessian[##] & @@@ m
{{{4, 2, 1}, {2, 0, 0}, {1, 0, 54}}, {{8, 4, 1}, {4, 0, 0}, {1, 0,
36}}, {{10, 18, 1}, {18, 0, 0}, {1, 0, 54}}, {{6, 6, 1}, {6, 0,
0}, {1, 0, 6}}, {{10, 12, 1}, {12, 0, 0}, {1, 0, 18}}}

det = Det[##] & /@ h
{-216, -576, -17496, -216, -2592}

• hessian @@@ m and Det /@ h suffice. – J. M.'s ennui Oct 20 '15 at 13:07
• @J.M. Thanks for the hint. – user31001 Oct 20 '15 at 13:29