unconstrained optimization Determine the maximum and minimum points and saddle points (if any) for each of the following functions a) g(x.y) = 2x- - 2xy + y3 + 5x - 3x b) u(x. y) = 4x + 2y x34 Mi Solve the following unconstrained optimization application exercises. 1. Determine the maximum utility if the production function is P = 20 - x- + 10x - 2y + 5y, the prices of inputs "x" and "'y" are 2 and 1, respectively, and the unit price of product P is 5. 2. A single grocery store in a rural community sells two brands of orange juice. The local brand has a cost of $ 0.20 per container and a foreign brand obtains at a cost of $ 0.40 per container. The store owner estimates that if the local brand sells "x" cents per container and the foreign brand sells a "'y" cents per container, then, approximately every day they sell 70-5x + 4y containers of the local brand, and 80 + 6x-7y of the foreign brand. At what prices should you sell local and foreign juice to maximize profit from the sale of juice? 3. A company produces "'x' units of commodity A and "y" units of merchandise B. All units can be sold ar p = 100 - x currency units (CU) per unit of A and q = 100 - y for B. The cost of producing these units is given by the joint cost function C(x, y) - x2. -a txy + 1 . what value must "x" and "y" have to maximize utility? Constrained optimization a) z = 3x- + 4y" - xy subject to: x + y = 21 b) z = -10x- +y subject to:x - y + 18 = 0