Question: With quantities x and y of two raw materials available, Q = x½y½ thousand items can be produced at a cost of C = 2x + y thousand dollars. Using the following steps, find the minimum cost to produce 1 thousand items.
(a) Graph x½y½ = 1. On the same axes, graph 2x + y = 2, 2x + y = 3, and 2x + y = 4.
(b) Explain why the minimum cost occurs at a point at which a cost line is tangent to the production curve Q = 1.
(c) Using your answer to part (b) and implicit differentiation to find the slope of the curve, find the minimum cost to meet this production level.