Problem 1. Find the maximum and the minimum distance from the origin to the ellipse
x^{2} + xy + y^{2} = 3.
Hints: (i) Use x^{2} + y^{2} 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 y^{2} + z^{2} = 1 and xz = 3.
Problem 3. (a) Maximize f (x, y) = x^{2 }+ y^{2} 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.