Constrained optimization exploratory activity

Solving constrained optimization problems with Lagrange multipliers:

Part 4: Analytical solution

(1) Solve the constrained optimization problem (*) with paper and pencil using the method of Lagrange multipliers. Organize your work as a clear, linear presentation with explanatory notes for each step.

(2) What if the constraint in (*) is revised to allow all points of the plane on or inside the sphere ?

(3) What if the planar constraint is dropped? Can you then solve the problem without calculus?

(4) Re-solve (*) without Lagrange multipliers by finding an explicit parameterization for the constraint curve.

(5) Which of the two solution methods for (*) (Lagrange multipliers vs. explicit parameterization of the constraint curve) do you think is more flexible in general? Why?

(6) Find the maximum and minimum values of subject to the constraints and .

(7) Can you identify any feature(s) of optimization problem (6) not in evidence in (*)?

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