Linear Algebra and Quadratic Forms

[Wint]

Homework Statement

For the quadratic form x²-2xy+2yz+z²:

a) Find a symmetric matrix that allows the quadratic form to be written as x^TAx.
b) Determine if the critical point at the origin is a minimum, maximum, or neither.
c) Find the points for which the quadratic form achieves its minimum and maximum values on the sphere x²+y²+z²=1.

Homework Equations

Unsure, I'm having troubles with part c and I think my notes may be missing something. I have written down here that:

x = Qy (x,y are matrices, Q is an orthonormal matrix) but I'm not sure how to use that. I have written next to it that the maximum is at x=q₁ and the minimum is at x=q_n where n is the dimension of the matrix and q is the column in Q, but I'm not sure what that means.

The Attempt at a Solution

I have figured out a and b easily enough:
The matrix is

The eigenvalues are 2, 1, and -1, which tells me that the critical point at the origin is neither a maximum or a minimum.

Which brings us to part c. Really I'm just trying to find anything that explains further how we find these maximum and minimum values, and any guidance would be much appreciated.

 

[Wint]

Nevermind, I found the section in my book that discusses this, and I've figured it out. Seems like that usually happens right after I ask the question.