If the equation a_nx^n+a_n-1x^n-1++a_1x=0 (a_10, n2) has a positive root x=, then the equa…

Question

If the equation $a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x=0$ ($a_1
e0$, $n\ge2$) has a positive root $x=\alpha$, then the equation $na_nx^{n-1}+(n-1)a_{n-1}x^{n-2}+\cdots+a_1=0$ has a positive root, which is

Accepted Answer

Let $f(x)=a_nx^n+\cdots+a_1x$. $f(0)=0=f(\alpha)$ (given root). By Rolle's theorem, $f'(x)=na_nx^{n-1}+\cdots+a_1=0$ has a root strictly between $0$ and $\alpha$, i.e. smaller than $\alpha$.

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