To solve this problem, we can use the method of Lagrange multipliers to find the minimum value of $r(x_1, x_2, x_3)$ subject to the constraint $q(x_1, x_2, x_3) = 1$.

Part (a): Finding the Minimum of $r(x_1, x_2, x_3)$

1. Define the functions: $r(x_1, x_2, x_3) = x_1^2 + x_2^2 + x_3^2$ $q(x_1, x_2, x_3) = x_1^2 + 3x_2^2 + x_3^2 + 2x_1x_2 - 2x_1x_3 - 2x_2x_3 = 1$

2. Set up the Lagrangian: Define the Lagrangian $\mathcal{L}$ as follows: $\mathcal{L}(x_1, x_2, x_3, \lambda) = r(x_1, x_2, x_3) - \lambda (q(x_1, x_2, x_3) - 1)$

3. Find the partial derivatives and set them to zero: We need to compute $\frac{\partial \mathcal{L}}{\partial x_1}$, $\frac{\partial \mathcal{L}}{\partial x_2}$, $\frac{\partial \mathcal{L}}{\partial x_3}$, and $\frac{\partial \mathcal{L}}{\partial \lambda}$ and set each equal to zero.

4. Solve the system of equations: Solving these equations simultaneously will yield the values of $x_1$, $x_2$, $x_3$, and $\lambda$ that satisfy the conditions for a minimum under the constraint.

Part (b): Maximum Value of $r(x_1, x_2, x_3)$

To determine if $r(x_1, x_2, x_3)$ has a maximum, we need to analyze whether the function $r(x_1, x_2, x_3) = x_1^2 + x_2^2 + x_3^2$ is bounded above on the set where $q(x_1, x_2, x_3) = 1$.

Given that $r(x_1, x_2, x_3)$ represents a sum of squares, if we examine values for which $q(x_1, x_2, x_3) = 1$, we can find that the constraint may allow values of $r(x_1, x_2, x_3)$ to grow without bound, indicating that there is no finite maximum.

Thus, the answers are:

1. (a) The minimum value of $r(x_1, x_2, x_3)$ under the constraint can be found by solving the system from part (a).
2. (b) There is no maximum value for $r(x_1, x_2, x_3)$ on the set where $q(x_1, x_2, x_3) = 1$.

Would you like more detailed steps or a further breakdown of any specific part?

Here are related questions that may help deepen understanding:

1. What are Lagrange multipliers, and how are they used in optimization problems with constraints?
2. How do we interpret the constraint $q(x_1, x_2, x_3) = 1$ geometrically?
3. Why does the lack of a finite upper bound imply there is no maximum for $r(x_1, x_2, x_3)$?
4. Can the method of Lagrange multipliers be used to find saddle points as well as minima and maxima?
5. What does the result of part (b) suggest about the geometry of the set where $q(x_1, x_2, x_3) = 1$?

Tip: When dealing with quadratic constraints, check if the constraint form represents an unbounded set, as this often hints at an absence of finite extrema for unconstrained variables.