Section 14.8: Lagrange Multipliers

Consider the problem of maximizing the function \(f(x, y) = 2x + 3y\) subject to the constraint \(\sqrt{x} + \sqrt{y} = 5\). (a) Try using Lagrange multipliers to solve the problem. (b) Does \(f(25, 0)\) give a larger value than the one in part (a)? (c) Solve the problem by graphing the constraint curve and several level curves of \(f\). (d) Explain why the method of Lagrange multipliers fails to solve the problem. (e) What is the significance of \(f(9, 4)\)?