Application of derivatives?

**fardeen_gen** · May 11th 2009, 09:50 PM

Given that $\sum_{k = 0}^{n - 1} a_{k} = 0$ where $a_{k} \in \mathbb{R}\ \forall\ k = 0, 1, 2,\mbox{....},(n - 1)$ .

Show that $a_{i_{0}}\cdot n\cdot x^{n - 1} + a_{i_{1}}\cdot (n - 1)\cdot x^{n - 2} + ... + a_{i_{n - 2}}\cdot 2x + a_{i_{n - 1}} = 0$ has at least one real root in $(-3, 3)$ for any permutation $(a_{i_{0}}, a_{i_{1}},\mbox{.....},a_{i_{n - 1}}\ \mbox{of}\ a_0,a_1,\mbox{.....},a_{n - 1})$ .

**NonCommAlg** · May 11th 2009, 10:12 PM

Originally Posted by fardeen_gen

Given that $\sum_{k = 0}^{n - 1} a_{k} = 0$ where $a_{k} \in \mathbb{R}\ \forall\ k = 0, 1, 2,\mbox{....},(n - 1)$ .

Show that $a_{i_{0}}\cdot n\cdot x^{n - 1} + a_{i_{1}}\cdot (n - 1)\cdot x^{n - 2} + ... + a_{i_{n - 2}}\cdot 2x + a_{i_{n - 1}} = 0$ has at least one real root in $(-3, 3)$ for any permutation $(a_{i_{0}}, a_{i_{1}},\mbox{.....},a_{i_{n - 1}}\ \mbox{of}\ a_0,a_1,\mbox{.....},a_{n - 1})$ .

trivial!

let $f(x)=a_{i_0}x^n + a_{i_1}x^{n-1} + \cdots + a_{i_{n-1}}x.$ then $f(0)=f(1)=0$ and thus by Rolle's theorem $f'(x)=0$ has a root in the interval $(0,1).$

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