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Math Help - Don't understand use of auxiliary function in proving theorems

  1. #1
    Member ssadi's Avatar
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    Don't understand use of auxiliary function in proving theorems

    How was the auxiliary function to be used in proving the theorem below determined? I don't understand why this function was chosen, and how the author arrived at the decision of choosing this function? This is the first time I came across the use of auxiliary functions in proving a theorem, so I am a bit confused whether the function is chosen by pure intuition or is there a method behind this.

    Thanks a lot for reading.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by ssadi View Post
    ... so I am a bit confused whether the function is chosen by pure intuition or is there a method behind this.
    The important thing is that F works and you understand why works. All the same, in this case, analyze:

    1.- G(x)=f(x)-f(a) translates vertically the graph of f in such a way that G(a)=0 .

    2.- The equation of the straight line containing the endpoints of the G graph is:

    y=\dfrac{f(b)-f(a)}{b-a}(x-a)

    Regards.

    Fernando Revilla
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