3. Using the method of Lagrange multipliers, find the global maximum and minimum for the function f(x, y) = x²+2y2 subject to the constraint x² + y² = 1.

23. Consider a simple economy with just two industries: farming and manufacturing. Farming consumes 1/2 of the food and 1/3 of the manufactured goods. Manufacturing consumes 1/2 of the food and 2/3 of the manufactured goods. Assuming the economy is closed and in equilibrium, find the relative outputs of the farming and manufacturing industries.

Use Lagrange multipliers to find the maximum and minimum values of the function f(x,y,z) = x² + y² +2² subject to the constraint x²+y²+2+xy=12. You may assume ftakes both a maximum and minimum value subject to the constraint.

Use Lagrange multipliers to find the maximum and minimum values of f(x, y) = xy subject to the constraint that 4x² + y² = 8.