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Consider the problem of minimizing x2 y2 subject to x 2y a

Question: (a) Consider the problem of minimizing x² +y² subject to x +2y = a (where a is a constant). Solve the problem by transforming it into an unconstrained optimization problem with one variable.

(b) Show that the Lagrange method leads to the same solution. Verify (2) in this case.

(c) Explain the solution by studying the level curves of f (x, y) = x² +y² and the graph of the straight line x + 2y = a. Can you give a geometric interpretation of the problem? Does the corresponding maximization problem have a solution?

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