Max-min Problems

Max-min problems are unique to calculus. As the name implies, a variable is maximized or minimized in terms of another variable. A typical problem would ask the question:

"What is the maximum volume of a cylindrical container that can be made from a given amount of material?" The volume of the container is the variable to be maximized while the surface area of the container is limited by the amount of material allowed. In this example an equation for the volume ( V = ... ) is the defining equation. It defines the variable to be maximized, the volume, in terms of the dimensions of the container. The specification of a certain amount of material for the container is called the constraint equation. It relates the variables in the defining equation so the defining equation can be written in terms of one variable. This all becomes much clearer after a couple of problems.

Once the defining equation is written in terms of one variable it is differentiated to find where the slope is zero. Where the slope of this curve is zero, the curve is at a maximum or a minimum. The value of the second derivative tells whether that point is a maximum or a minimum. Finding the points where the slope is zero and then identifying those points as either maximum Ç, or minimum È, has already been done in the graphing section. Max-min problems use much the same analysis techniques as with graphing.

Writing the defining equation is usually relatively easy. The hard part of max-min problems is finding the constraint equation and then doing the algebra so as to get the defining equation written in terms of one, other than the one to be maximized or minimized, variable and in as simple a form as possible.

There are very few max-min problems where the defining equation is written directly in terms of one variable. They are seen rarely on tests. They are considered too easy! Let's slowly go through a couple of max-min problems before setting down guidelines for working the problems and going on to the more challenging problems. Learn the procedure and max-min problems are not difficult.

Maxmin02 - Enclosing a rectangular area.

Guidelines for Max-Min Problems

1. Draw a diagram to help visualize the problem.
2. Write down the defining equation.
3. Tie the two variables in the defining equation together with a constraint equation.
4. Write the defining equation in terms of one variable.
5. Take the first and second derivatives to find maxima and minima.
6. Go back to the constraint equation and find all the quantities desired in the problem.

Maxmin03 - The strength of a rectangular wooden beam.

Maxmin04 - Fencing three sides of an area.

Maxmin05 - Yield of a field of crops.

Maxmin06 - Minimum cost to construct a cylinder.

Maxmin07 - Dimensions of a bricklike box.