Problem 1. Find the maximum and the minimum distance from the origin to the ellipse

x² + xy + y² = 3.

Hints: (i) Use x² + y² as your objective function; (ii) You can assume that the constraint qualification condition and the second order conditions are satisfied in this problem, as well as in problems 2 and 3.

Problem 2. Maximize f (x, y, z) = yz + xz subject to y² + z² = 1 and xz = 3.

Problem 3. (a) Maximize f (x, y) = x² + y² subject to 2x + y ≤ 2, x ≥ 0 and y ≥ 0.

(b) Use the Envelope Theorem to estimate the maximal value of the objective function in part

(a) when the first constraint is changed to 2x+ 9/8y ≤ 2, the second constraint is changed to x ≥ 0.1,and the third to y ≥ -0.1.