Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
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.