Polynomial

**token22** · May 19th 2011, 04:48 AM

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

**Opalg** · May 19th 2011, 08:36 AM

Originally Posted by token22

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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