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00test2

B does u attain a maximum on the set b xy y 0 px y 3

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b ) Does u attain a maximum on the set B = { ( x,y ) : y 0, px + y 3 } ? Why? Answer: This is a trickier problem. As was the case with the cost minimization problem done in class, we must simplify the problem before solving it. First, note that u is increasing in both x and y . It follows that any maximum must occur along the constraint px + y = 3. So x = (3 - y ) /p . Substituting into u , we reduce our problem to maximizing (3 - y ) /p + y over the set y 0. Note that the maximum must be at least 3 /p , which is the value taken at y = 0. It is easy to see that lim y →∞ (3 - y ) /p + y = -∞ , so there is some b with (3 - y ) /p + y < 3 /p for y > b . We can now remove any y > b from the set we are maximizing over, and focus on whether (3 - y ) /p + y can be maximized over [0 ,b ]. But now we are maximizing a continuous function over a compact set, and Weierstrass’s Theorem tells us that there is a maximum. This is also a maximum for the original problem.
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