, we reduce our problem to maximizing (3-y)/p+√yoverthe sety≥0. Note that the maximum must be at least 3/p, which is the value taken aty= 0.It is easy to see that limy→∞(3-y)/p+√y=-∞, so there is somebwith (3-y)/p+√y <3/pfory > b. We can now remove anyy > bfrom the set we are maximizing over, and focus onwhether (3-y)/p+√ycan be maximized over [0, b]. But now we are maximizing a continuousfunction over a compact set, and Weierstrass’s Theorem tells us that there is a maximum. Thisis also a maximum for the original problem.
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