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Math Help - Find Function

  1. #1
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    Find Function

    Hello, Need your help

    Find all function f: \mathbb{R} \longrightarrow [-1,1] verifying \forall x \in  \mathbb{R} , f(2x)=2\left(f(x)\right)^2 -1 , such that f(0)=1 and \lim_{x\rightarrow 0}\frac{1-f(x)}{x^2} = a , a \in  \mathbb{R}^{+}
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  2. #2
    MHF Contributor Calculus26's Avatar
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    Actually cos[(2a^1/2)x] will work for any a in R+
    Last edited by Calculus26; April 16th 2009 at 04:46 PM.
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  3. #3
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    hello,

    hi Calculus26 , I know it's \cos(x\sqrt{2a}) but I need proof ...
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  4. #4
    MHF Contributor Calculus26's Avatar
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    If you need to verify:

    f(0) =1 is obvious

    using the double angle formula cos(2t) = 2cos^2(t)-1 for t = (2a)^1/2 x

    which is precisely 2f(x)^2 -1

    double application of L'Hopital's Rule verifies the limit

    Now if you need to come up with cos((2a)^1/2 x) from first principles then that's another story. Do we have any other conditions such as differentiability ?

    My thought process was simply

    1. The condition f:R - [-1,1] led me to cosines and sines

    2. f(0) =1 narrowed this to cos(bx)

    3. the limit led to b^2 = 2a

    Let me know if this not sufficient and I'll consider it further
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  5. #5
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    Integers identity

    Hello,


    thanks claculus26
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