Consider two nonnegative numbers x and y such that

Question

x + y = 12.
Find the values of x and y that optimize the following quantity.
xy
Maximize the quantity.
{x, y} =




Minimize the quantity. (Order your answers from smallest to largest x, then from smallest to largest y.)
{x, y}
=




{x, y}
=

Answer 1

xy = x(12-x)
that is a parabola with vertex at x=6

This is just an illustration that the rectangle with the largest area for a given perimeter is a square.

clearly the minimum is achieved when the rectangle is just a line of length 12 and area zero.