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Quadratic problem. any other method?

  1. Oct 21, 2012 #1
    quadratic problem. any other method??

    1. Thie problem statement, all variables and given/known data
    I was solving this question paper and came across the second problem :
    http://olympiads.hbcse.tifr.res.in/uploads/rmo-2010 [Broken]

    I wanted to solve this one by calculus.

    I want to know that, does in problem like this, we may use:
    If P(x)=Q(x)
    Then, P'(x)=Q'(x), P''(x)=Q''(x) etc.?

    I know this equality may not be true for other functions, but is it applicable for this specific problem?
     
    Last edited by a moderator: May 6, 2017
  2. jcsd
  3. Oct 21, 2012 #2

    ehild

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    Re: quadratic problem. any other method??

    You can not assume that the functions are identical, they just happen to return the same value at a specific x.
    See the picture: although all function have the same value at a certain x, their derivatives are different.

    That is an other thing that these special functions can have point(s) in common only when they are identical.

    ehild
     

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    Last edited: Oct 21, 2012
  4. Oct 21, 2012 #3
    Re: quadratic problem. any other method??

    Thanks for reply. Yes I understand what you meant. Can you suggest some good method for this problem?
    Let P1(x) = ax2 − bx − c,
    P2(x) = bx2 − cx − a,
    P3(x) = cx2 − ax − b
    be three quadratic
    polynomials where a, b, c are non-zero real numbers. Suppose there exists a real number
    α such that P1(α) = P2(α) = P3(α). Prove that a = b = c.
     
  5. Oct 21, 2012 #4

    ehild

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    Re: quadratic problem. any other method??

    Sorry, I can not find any other method different from that given. It is a nice and simple method, (except a mistake in sign, but it does not matter) what is your problem with it?

    ehild
     
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