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Agentjamesbond007

  • 5 years ago

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

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  1. myininaya
    • 5 years ago
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    \[xy=100\] let's call the Sum S S=x+y

  2. myininaya
    • 5 years ago
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    now solve xy=100 for either x or y

  3. myininaya
    • 5 years ago
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    \[y=\frac{100}{x}\]

  4. anonymous
    • 5 years ago
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    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
    • 5 years ago
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    \[Sum=x+\frac{100}{x}\]

  6. myininaya
    • 5 years ago
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    now do S'

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

  8. anonymous
    • 5 years ago
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    Noooooo derivativeee pleaseeeeee!!!

  9. anonymous
    • 5 years ago
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    Simple inequalities would do just fine.

  10. anonymous
    • 5 years ago
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    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
    • 5 years ago
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    If the product of the two variables is constant, the sum of the two variables will be least when they are equal.

  12. anonymous
    • 5 years ago
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    @FoolForMath: but how to prove that rigorously without calculus / derivatives?

  13. anonymous
    • 5 years ago
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    @Broken Fixer:AM-GM inequality.

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

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