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1. A rectangular bird sanctuary is being created with one side along a straight riverbank. The remaining three sides are to be enclosed with a protective fence. If there are 12 km of fence available, find the dimension of the rectangle to maximize the area of the sanctuary.
2. The rectangular bird sanctuary with one side along a straight river is to be constructed so that it contains 8 km2 of area. Find the dimensions of the rectangle to minimize the amount of fence necessary to enclose the remaining three sides.
3. Find two positive real numbers such that the sum of the first number squared and the second number is 48 and their product is a maximum.
4. Find two positive real numbers such that they sum to 108 and the product of the first times the square of the second is a maximum.
5. A wire of length 12 m is divided into two pieces and the pieces are bent into a square and a circle. How should this be done in order to minimize the sum of their areas?
6. Find the positive number x such that the sum of x and its reciprocal is as small as possible. Does this problem require optimization over an open interval or a closed interval?
7. Find two positive real numbers such that they add to 40 and their product is as large as possible.
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The specified 7 problems are of Maxima-Minima problems. Some of them are directly based on the theory. They are solved by applying the theories of maxima and minima of a function of single variable. Problem no 3,4,6,7 are of this type. While the problems no 1,2 and 5 are practical problems which are needed to be converted into a mathematical model and then they are solved applying the theory.
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