step by step solution of the given.

Question

1.) Suppose that g(x,y,z)=x³y²z^-1.

Determine its Hessian H(g) at the point (1,1,1).

Answer 1

[g_xx g_xy g_xz]

H(g) = [g_yx g_yy g_yz]

[g_zx g_zy g_zz]

g(x,y,z) = x³y²/z

g_x = 3x²y²/z

g_y = 2x³y/z

g_z = -x³y²/z²

g_xx=6xy²/z g_xx(1,1,1)=6

g_xy=6x²y²/z g_xy(1,1,1)=6

g_xz=-2x²y²/z² g_xz(1,1,1)=-2

g_yx=6x²y/z g_yx(1,1,1)=6

g_yy=2x³/z g_yy(1,1,1)=2

g_yz=-2x³y/z² g_yz(1,1,1)=-2

g_zx=-3x²y²/z² g_zx(1,1,1) = -3

g_zy=-2x³y/z² g_zy(1,1,1) = -2

g_zz=-2x³y²/z³ g_zz(1,1,1) = -2

[6 6 -2]

H(g(1,1,1)) = [6 2 -2]

[-3 -2 -2]