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Homework Help: Proving inequality

  1. Sep 18, 2009 #1
    1. The problem statement, all variables and given/known data

    Let x1, x2, y1, y2 be arbitrary non-zero constants. Let

    x = x1 / root((x1)^2+(x2)^2)

    y = y1 / root((y1)^2+(y2)^2)

    Show that

    (2(x1)(y1))/(root(x1)^2+(x2)^2)root(y1)^2+(y2)^2)) =< (x1)^2/(x1)^2+(x2)^2 + (y1)^2/(y1)^2+(y2)^2

    2. Relevant equations

    3. The attempt at a solution

    Well, I get 2xy =< x^2 + y^2 by replacing that whole complex equation above.

    Then it becomes similar to that other post I made...

    making a contradictory statement:

    2xy >= x^2 + y^2

    0 >= x^2 - 2xy + y^2

    0 >= (x-y)^2

    but (x-y)^2 must be either 0 or a positive integer

    0 =< (x-y)^2

    0 =< x^2 - 2xy + y^2

    2xy =< x^2 + y^2

    and if I replace x and y, I get this again:

    (2(x1)(y1))/(root(x1)^2+(x2)^2)root(y1)^2+(y2)^2)) =< (x1)^2/(x1)^2+(x2)^2 + (y1)^2/(y1)^2+(y2)^2

    Am I doing this right? It doesn't really seem like it...
  2. jcsd
  3. Sep 18, 2009 #2
    This question is just a very exaggerated way of saying a very simple thing

    Arithmetic Mean of two numbers is greater than the Geometric Mean, i.e

    (a+b)/2 >= sqrt(ab) always, for a,b > 0

    just substitute(as you already have) a = x^2 and b = y^2 and u get the same inequality back

    your method is correct
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