3. Inequality-constrained optimization:

(a) Find the values for x, x2 which minimize the function f(x) = x - x subject to the constraints x 0, x 0, x + x 1. (Hint: recall that to minimize f(x), it suffices to maximize -f(x)).

(b) In the consumer's utility maximization pro- blem, treated in some detail in section 8.3, it is assumed that the budget constraint will bind. A more general formulation (using the same notation) would be as follows: Maximize U(x) subject to Pix, Y. Show that the optimal solution requires that the budget constraint must bind. (Hint: set up the Lagrangian and explore the necessary condi- tions; show that the multiplier on the constraint must be positive, and hence, using the comple- mentary slackness condition, that the constraint must bind.)