Unconstrained Optimization, Lagrange problem, Least Squares

Image transcriptions

Production function of a company is given by f(X,y) = «I; + J; wherex is labor and y is capital. Cost of labor is 2 currency units, while the cost of capital is 1 currency unit. What are the minimal cost to produce 10 units of the product? What is the change in the costs, if the production volume would increase by 50 %? The total weekly revenue (in dollars] of Country Workshop associated with manufacturing and selling their rolltop desks is given by the function R(x, y) =—0,2:c2 —0,25y2 —0,2x5v +200x+ 160 y where x denotes the number of ﬁnished units and )1 denotes the number of unﬁnished units manufactured and sold each week. The total weekly cost attributable to the manufacture of these desks is given by TC(x, y) = 100x + 70y + 4000 dollars. Determine how many ﬁnished units and how many unﬁnished units the company should manufacture and sell each week in order to maximize its weekly proﬁt? What is the maximum proﬁt realizable?

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