Suppose that g(x,y,z)= x^3y^2z^-1 determine the hessian h(g) at the point (1,1,1)

Question

I actually have an initial solution to this, I just got stuck and I do not know if what I did was right

g(x,y,z) =x³y²z^-1

Fx=3x²y² z^-1

Fy=x³2yz^-1

Fz=x³y²-z^-1

Then the second partial derivative

Fxx= 6xy²z^-1 Fxy= 3x²2yz^-1 Fxz= 3x²y²-z^-2

Fyx= 3x²2yz^-1 Fyy= x³2z^-1 Fyz= x³2y-z^-2

Fzx= 3x²y²-z^-2 Fzy= x³2y-z^-2 Fzz= x³y²2z^-3

Then I evaluate it by 1 because of the point (1,1,1)

Fxx= 6xy²z^-1 =6 Fxy= 3x²2yz^-1 = 6 Fxz= 3x²y²-z^-2 = 3

Fyx= 3x²2yz^-1 = 6 Fyy= x³2z^-1 = 2 Fyz= x³2y-z^-2 = 2

Fzx= 3x²y²-z^-2 = 3 Fzy= x³2y-z^-2 = 2 Fzz= x³y²2z^-3 = 2

We got stuck after this matrix, we don't know whats should we do next? should we find the determinant of the matrix? How are we supposed to know if it is a saddle point, minimum, or maxima?

The lecture links have different types of given and examples they equate it first to 0 to find the critical points then proceed to the hessian matrix but we can't seem to apply it here because it is a different type of given. I also do not know if we can apply the test to find if it is a minimum, maxima, or saddle point

Answer 1

The mistake is you wrote the minus as subtraction in the derivative and not as a coefficient.

As you said, you need the determinant of the Hessian matrix.

F_y

F_z=-x³y²z^-2

F_xz= F_zx = -3x²y²z^-2 giving a value of -3

F_yz = F_zy = -2x³yz^-1 giving a value of -2

F_zz =2x³y²z^-3 giving a value of 2

The matrix comes from putting the correct values in the matrix.

Now you need to determine the eigenvalues λ such that F-λI = 0

if all the λ values are positive, you have a local minimum

if all the λ values are negative, you have a local maximum

if λ take both positive and negative values, you have a saddle point.