If the equation $a_n x^n+a_{n-1} x^{n-1}+\ldots \ldots+a_1 x=0, a_1 \neq 0, n \geq 2$, has a positive root…

If the equation $a_n x^n+a_{n-1} x^{n-1}+\ldots \ldots+a_1 x=0, a_1 \neq 0, n \geq 2$, has a positive root $x=\alpha$, then the equation $n a_n x^{n-1}+(n-1) a_{n-1} x^{n-2}+\ldots . .+a_1=0$ has a positive root, which is
  1. greater than $\alpha$
  2. smaller than $\alpha$
  3. greater than or equal to $\alpha$
  4. equal to $\alpha$

Solution

$f(0)=0, f(\alpha)=0$ $\Rightarrow f^{\prime}(k)=0$ for some $k \in(0, \alpha)$

Asked in: JEE Main 2005

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