I have the following function:
$x^2+6xy+y^2-3yz+4z^2-10x-5y-21z$
and I am trying to find the critical points to determine if they are max, min, or saddle points. For some reason, I am struggling with finding the critical points, and am wondering if I am missing something obvious or going about this all wrong.
My understanding of the procedure for solving something like this is:
1) Take partial derivatives
This gives:
$\partial/dx = 2x + 6y + 0z -10$
$\partial/dy = 6x + 2y - 3z -5$
$\partial/dz = 0x - 3y +8z -21$
2) Set partial derivatives equal to zero
3) Put partial derivative equations into matrix form and solve (and then 4) set up Hessian and evaluate principal minors to determine if the point is a minimum, a maximum, or a saddle point, but I'm stuck on #3)
$$ \left[ \begin{array}{ccc|c} 2&6&0&10\\ 6&2&-3&5\\ 0&-3&8&21 \end{array} \right] $$
This gives $x = 2, y = 1, z = 3$
Plugging those into the partials, I get
$\partial/dx = 2(2) + 6(1) + 0(3) -10 = 0$ check
$\partial/dy = 6(2) + 2(1) - 3(3) -5 = 0$ check
$\partial/dz = 0(2) - 3(1) +8(3) -21 = 0$ check
So what is the problem? The book gives the following answer:
$(-369/137, -14/137,29/137)$
Plugging that back in doesn't seem to come close to satisfying the criteria that the partial derivatives = 0. For the life of me, I have no idea where it came from, whether it is an error, or something else. I don't feel confident enough in these problems to think I am right, but I am struggling to figure out where I went wrong.