Question

Find two numbers whose difference is 68 and whose product is a minimum. (smaller number) (larger number)

Answer 1

Let P be the product, x be the first number and y be the second number
therefore:
x-y=68........i
P=xy....ii
from i
x=68+y
hence
P=y(68+y)=y^2+8y
P=y^2+8y
The minimum product will occur at point where:
dP/dy=0
thus
dP/dy=2y+8=0
thus
y=-4
thus the value of x will be:
x=68-4=64
thus the product will be:
-4*64=-256 sq. units