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In this situation, we must check what happens to the function as x approaches positive and negative infinity. By inspection, it becomes clear that as x approaches positive infinity, f also approaches positive infinity. Thus, the function grows without bound, and there is no absolute maximum.

Constrained Optimization

A builder needs to make a box with a square bottom and rectangular sides. The box has no top. If the material for the sides cost $2 per square foot, and the material for the bottom costs $4 per square foot, what is the largest volume box that the builder can make with $20?

This problem is known as a "constrained optimization" problem. The procedure for solving this sort of problem is ultimately similar to the procedure described above for optimizing functions of one variable. However, some work is required to transform this word problem into a function of one variable. The first three steps below describe this process.


Step One: Identify the objective function and express it in terms of the relevant variables.

The objective function represents the quantity that is ultimately going to be maximized or minimized. In this case, the quantity of interest is the volume of the box, and it needs to be maximized. The relevant variables here are the dimensions of the box. It is often useful to draw a diagram:

Let x be the both the length and width of the square bottom of the box.
Let y be the height of the sides of the box.

Expressing the volume in terms of the relevant variables generates the objective function: V = x2y. This quantity must be maximized.

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