(b) The optimal solution to the original problem, in the variables x1,x2𝑥1,𝑥2 and x3𝑥3, is given by:

Determine the maximum and minimum values of quadratic form 𝑄(𝑥) subject to the constraint 𝑥𝑇𝑥=1.𝑄(𝑥)=6𝑥12+6𝑥22−10𝑥1𝑥2

A consumer spends an amount 𝑚𝑚 to buy 𝑥𝑥 units of one good at the price of 6 per unit and 𝑥𝑥 units of a different good at the price of 10 per unit. Here 𝑚𝑚 is positive and suppose that 8 < 𝑚𝑚 < 40. The consumer’s utility function is 𝑈(𝑥, y) = 𝑥y + y^2+ 2𝑥 + 2y, so that her problem is: Maximize 𝑥y+y^2 +2x+2y subject to 6𝑥+10𝑥y=𝑚 (a) Find the optimal quantities 𝑥∗ and y∗ and the Lagrange multiplier, all of them as functions of 𝑚. (b) Write down the maximum value of the utility function as a function of m. (c) Find the derivative of the above utility function for m=20 (d) What are the solutions for 𝑥∗ and y∗ if (i) 𝑚 ≤ 8? (ii) 𝑚 ≥ 40?

Optimality. Consider the following optimization problemminimize 12 xT P x + qT x + rsubject to −1 ≤ xi ≤ 1, i = 1, 2, 3whereP = 26 24 −424 34 12−4 12 24 , q =−44−2926 , r = 1.(a) Prove that x? = (1, 0.5, −1) is optimal.(b) Are any of the constraints inactive at x?? If so, which one(s)?

The Optimization Problem Involves

optimization procedure