Use Lagrange multipliers to find the maxi- mum value of the function f (x, y, z) = x + 2y subject to the constraints x + y + z = 8, y² + 2² = 36. 1. 6 – 6V6 - 2. 8 +7V6 3. 8 + 6V2 4. 8 – 6V2 - 5. 6 — 6у2 -

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.

Find the maximum of V = xyz subject to the constraint 6x+4y+3z-24=0.

Draw the box and show how to get the edge lenght and surface area equation from it