Math

Consider two nonnegative numbers x and y such that
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}
=

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  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.

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