Calculate the family of optimization models

Question

Assignment:

Consider the family of optimization models:

         max x₁ + x₂

         s.t.  x₁ + x₂ ≤ t

 (P_t)   x²₁ + x²₂ ≤ 1/1 + t²

         x₁ ≥ 0, x₂ ≥ 0,

where t is a parameter (it is not a variable)

1. Solve (P_t) when t = 1.

2. For what values of t is the solution (1,0) feasible for (P_t)?

3. For what values of t is (P_t) infeasible?

4. For what values of t is the feasible region of (P_t) convex? (Technically, an empty set is convex)

5. For what values of t does (P_t) have a single optimal solution.

6. For what values of t is the solution (t, 0) optimal for (P_t)

7. For what values of t does (P_t) have an infinite number of solutions, but (t, 0) is not one of them. (It might be useful to observe that the equation at⁴  + bt² + c= 0 can be easily solved if you know how to solve ax² + bx + c = 0).