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Math Help - Integral Inequality

  1. #1
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    Integral Inequality

    I am trying to prove that the shortest distance between two points is a straight line. And it is almost complete, if I can use the following theorem.
    Assume,
    f(x),g(x) are Integratable on [a,b].
    And,
    \int_a^b [f(x)]^2 dx\leq \int_a^b [g(x)]^2 dx
    Then is it true that,
    \int_a^b |f(x)|dx\leq \int_a^b |g(x)|dx?

    And one more,
    0\leq \int_a^b f(x)dx\leq \int_a^b g(x)dx
    Und,
    0\leq \int_a^b a(x)dx\leq \int_a^b b(x)dx
    Then,
    0\leq \int_a^b f(x)a(x)dx\leq \int_a^b g(x)b(x)dx

    I tried the Cauchy-Swarthz inequality because this is an inner product space but it does not help sufficiently.
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  2. #2
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    Hey PH. Here's a nice proof if you wanna check it out. Something you may like.

    Proof shortest distance between two point is a straight line
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  3. #3
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    Quote Originally Posted by galactus View Post
    Hey PH. Here's a nice proof if you wanna check it out. Something you may like.

    Proof shortest distance between two point is a straight line
    Well, I did it with my professor. The proof is so much nicer. This happens to be a Calculus of Variations problem. He told me the standard way is to define y "what you think is the answer is" in this case the line passing through the two points. And let y+\delta y be any other path taken, where \delta y is some smooth function. And then show the arc length of y+\delta y is more than the arc length of y. I was able to do it by the use of the inequalities above. And if someone says they are valid (and I think they are) I can post my solution.
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