Find two positive numbers whose product is 100 and whose sum is a minimum.

[please explain how to solve]

Question

Find two positive numbers whose product is 100 and whose sum is a minimum.

[please explain how to solve]

myininaya · Answer

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

myininaya · Answer

now solve xy=100 for either x or y

myininaya · Answer

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

anonymous · Answer

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.

myininaya · Answer

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

myininaya · Answer

now do S'

anonymous · Answer

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

anonymous · Answer

Noooooo derivativeee pleaseeeeee!!!

anonymous · Answer

Simple inequalities would do just fine.

anonymous · Answer

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.

anonymous · Answer

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

anonymous · Answer

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

anonymous · Answer

@Broken Fixer:AM-GM inequality.

anonymous · Answer

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