Consider the family of optimization models:

φ(t) = max tx1 − x2

s. t. x1 − x2 ≤ t

(pt) 0 ≤ x1 ≤ 1

0 ≤ x2 ≤ 1,

where t ∈ r is a parameter (it is not a variable).

1. use the graphical method to solve (pt) when t =0.5 (describe both an optimal solution (x1,x2) and the optimal value φ(.5)

2. determine all values of t (if any) for which (pt) has no feasible solution.

3. determine all values of t (if any) for which (pt) has more than one optimal solution.

4. argue that the feasible region of (pt) contains that of (pτ ) if t ≥ τ .

5. argue that the function φ(t) is an increasing function of t. (in other terms, you should argue that φ(t) ≥ φ(τ ) if t ≥ τ .) 6. (bonus) evaluate the derivative of φ(t) when (i) t ∈ (−1, 0), (ii) t ∈ (0, 1), and (iii) t ∈ (1, [infinity])