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Math Help - Taylor problem #1

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    MHF Contributor Also sprach Zarathustra's Avatar
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    Taylor problem #1

    Let n>=2 and p(x)=x^n + a1*x^(n-1) + ... + an .
    Prove that to equation p(x)=sin(x) have at most n roots.
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  2. #2
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    Quote Originally Posted by Also sprach Zarathustra View Post
    Let n>=2 and p(x)=x^n + a1*x^(n-1) + ... + an .
    Prove that to equation p(x)=sin(x) have at most n roots.
    Between each two zeros of f(x) = p(x)-\sin x there will be a zero of f'(x). So f(x) can have at most one more zero than f'(x). Then between each two zeros of f'(x) there will be a zero of f''(x), and so on.

    Now look at what happens when you differentiate f(x) n times.
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