Question

If the equation $a_n{x}^n+a_{n-1}{x}^{n-1}+.........+a_{1}x=0,a_1\ne 0,n\ge 2,$ has a positive root~ $x=\alpha$, then the equation $n{a}_n{x}^{n-1}+(n-1)a_{n-1}{x}^{n-2}+.......+a_1=0$ has a positive root, which is :

• A. Smaller than ∝
• B. Greater than ∝
• C. Equal to ∝
• D. Greater than or equal to $\propto$

Answer 1

The correct option is A Smaller than ∝

Let $f\left(x\right)=a_n{x}^n+a_{n-1}{x}^{n-1}+.....+a_{1}x$
$\therefore f\left(0\right)=0$
$f\left(\alpha \right)=0$
$\Rightarrow f^{\prime }\left(x\right)=0$ has least one root between $(0,\alpha )$
$\Rightarrow n{a}_n{x}^{n-1}+(n-1)a_{n-1}{x}^{n-1}+.....+a_1=0$ has a positive root smaller than $\alpha$.