Solve the problem maxmin f x y z x2 y2 z subject to gx y

Question

Question: (a) Solve the problem, max(min) f (x, y, z) = x² + y² + z subject to g(x, y, z) = x² + 2y² + 4z² = 1

(The graph of the constraint is the surface of an ellipsoid in 3, a closed and bounded set.)

(b) Suppose the constraint is changed to x² + 2y² + 4z² = 1.02. What is the approximate change in the maximum value of f (x, y, z)?