Download Find two nonnegative numbers whose sum is 9 and

PROBLEM 1 : Find two nonnegative numbers whose sum is 9 and so that the
product of one number and the square of the other number is a maximum.
SOLUTION
Let variables x and y represent two nonnegative numbers. The sum of the two
numbers is given to be
9=x+y,
so that
y=9-x.
We wish to MAXIMIZE the PRODUCT
P = x y2 .
However, before we differentiate the right-hand side, we will write it as a
function of x only. Substitute for y getting
P = x y2
= x ( 9-x)2 .
Now differentiate this equation using the product rule and chain rule, getting
P' = x (2) ( 9-x)(-1) + (1) ( 9-x)2
= ( 9-x) [ -2x + ( 9-x) ]
= ( 9-x) [ 9-3x ]
= ( 9-x) (3)[ 3-x ]
=0
for
x=9 or x=3 .
Note that since both x and y are nonnegative numbers and their sum is 9, it
follows that
. See the adjoining sign chart for P' .
If
x=3 and y=6 ,
then
P= 108 is the largest possible product.