Introduction

In general, each type of problem requires some quantity to be minimized or maximized. We assume that the quantity we want to optimize can be represented by a function. Once this function is determined, the problem can be reduced to the question of determining at what number the function assumes its maximum or minimum.

Even though each applied problem has its unique features, it is possible to outline in a rough way a procedure for obtaining a solution. This five-step procedure is:

Steps

Identify the quantity for which a maximum or a minimum value is to be found
Assign symbols to represent other variables in the problem. If possible, use an illustration to assist you.
Determine the relationships among these variables.
Express the quantity to be optimized as a function of one of these variables
Apply the test for maximum and minimum to this function.

From each corner of a square piece of sheet metal 18 centimeters on a side, remove a small square of side x centimeters and turn up the edges to form an open box. What should be the dimensions of the box so as to maximize the volume?

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