## Agentjamesbond007 4 years ago Find two positive numbers whose product is 100 and whose sum is a minimum. [please explain how to solve]

1. myininaya

$xy=100$ let's call the Sum S S=x+y

2. myininaya

now solve xy=100 for either x or y

3. myininaya

$y=\frac{100}{x}$

4. anonymous

xy = 100 or y = 100/x and we need to minimize F = x + y = x + 100/x now we are ready to take the derivative of F.

5. myininaya

$Sum=x+\frac{100}{x}$

6. myininaya

now do S'

7. anonymous

You can find the minimum of x + (100/x) using only Algebra 2 or Geometry if you're Calculus-impaired.

8. anonymous

9. anonymous

Simple inequalities would do just fine.

10. anonymous

Set S equal to the sum, x+(100/x), and solve that equation for x in terms of S. You'll get a parabola in x that has only one minimum at the Vertex of that parabola.

11. anonymous

If the product of the two variables is constant, the sum of the two variables will be least when they are equal.

12. anonymous

@FoolForMath: but how to prove that rigorously without calculus / derivatives?

13. anonymous

@Broken Fixer:AM-GM inequality.

14. anonymous

@Luis: if Sum = 100 then there is no way to attain the minimum product with positive reals (infimum is 0).