If they exist, find two numbers whose sum is 100and whose product is a minimum. If such two numbers do not exist, explain why.
Let A and B are the numbers which satisfy this criteria.
Then (A+B) = 100
If the product is P then;
P = A*B = A*(100-A) = 100A-A^2
Let consider (50-A)^2;
(50-A)^2 = 2500-2*50*A+A^2
(50-A)^2 = 2500-(100A-A^2)
(100A-A^2) = (50-A)^2-2500
P = -(50-A)^2+2500
(50-A)^2 is always >=0
So the function P is gradually decreasing.
When A=50 then P gives its maximum.
So there is no minmum to the product of the numbers.
When A=50 then B=50
No two numbers only one number satisfy the criterian even we have a maximum value.