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Math Help - Polynomial

  1. #1
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    Polynomial

    Hello i got this problem and i have no idea what to do:
    Among polynomials of third degree with coefficients in R find such f(x) for which is value {max}_{x \in  \left< -1,1  \right> } f(x) minimal.

    Thanks for any help
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  2. #2
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    Quote Originally Posted by token22 View Post
    Hello i got this problem and i have no idea what to do:
    Among polynomials of third degree with coefficients in R find such f(x) for which is value {max}_{x \in  \left< -1,1  \right> } f(x) minimal.
    You need some additional condition for this problem to make much sense. As it stands, the max value of the polynomial could be arbitrarily low. For example, if f(x) = x^3-10^{100} then the max value on the interval [1,1] is -10^{100}+1.

    My first thought was that the quantity to be minimised should be |f(x)|. But that is no better, because for example the function f(x) = \varepsilon x^3 has max absolute value as close as you like to 0, for sufficiently small \varepsilon .
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