# If they exist, find two numbers whose sum is 100and whose product is a minimum. If such two numbers do not exist, explain why.

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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.