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 + y~ + 5x - 3y bj u(x. y ) = 4x + 2y - xi + xy - v2 Solve the following unconstrained optimization application exercises. 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? .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 + xy 2 v + 12. what value must "x" and "y" have to maximize utility? Constrained optimization I by = = 3x- + 4y- - xy subject to: x ty - 21 - -10x-+ y- subject to:x - y + 18 = 0 Total, and differential derivative applications If Z = f (x, y) with x = g (t) and Y = h (t), how is the total derivative dz / dt is expressed? What does a total derivative measure us? What is the term (dz / dy . dy / dt) is telling us within the total derivative? The production of a certain plant is Q (x, y) = 0.08.x- + 0. 12xy + 0. 03y units per day, where 'x" is the number of hours of skilled labor used and "y" is the number of hours of unskilled labor. Today 80 hours of skilled labor and 100 hours of unskilled labor are employed every day. Use the total differential of Q to estimate the resulting change in production if 1 hour of skilled labor is added and 2 hours of unskilled labor are to be decreased. Interpret the result. 1 1 At a factory, output production is Q = 120K?LIunits, where K represents capital investment measured in units of $ 1,000 and L the size of the labor force measured in worker-hours. Currently, the capital investment is $ 250,000 and 400 worker-hours are employed. Apply the total differential of O to estimate the resulting change in output if capital investment increases by $ 500 and labor increases by 2 worker-hours