# Minimax problems.

For such problems, follow the flow chart:

Example 6.11.1   Find the dimensions of a rectangle with maximal area and a perimeter of meters.

Let be the length of the rectangle; the width verifies the relation: . The area of the rectangle is given by: .

Then: and if, and only if, .

As is a quadratic polynomial with negative main coefficient, the function has a maximum at the point . replacing in the definition of , we have: .

In order to have maximal area with a fixed perimeter, the rectangle should be a square.

Example 6.11.2   Consider a square plate whose side is 20 cm long. From each corner, cut a square in order to create a topless box by folding the sides upwards. Find the dimensions of the box for the volume to be maximal, then compute the maximal volume.

Denote by the side length of the small squares cut off at the corners of the plate. This provides a box whose volume is given by the function defined as follows:

Now we have:

and

 or

If , nothing is left and no box is built. Therefore .

Apply the second derivative test:

and . As this number is negative, we have a confirmation that for the volume of the box is maximal.

the maximal volume is given by:

 cm

Noah Dana-Picard 2007-12-28