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If the equation $a_{n} x^{n} + a_{n - 1} x^{n - 1} + \dots + a_{1} x = 0$ has a positive root $α,$ then the equation $n a_{n} x^{n - 1} + (n - 1) a_{n - 1} x^{n - 2} + \dots + a_{1} = 0$ has

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a positive root less than

α

a positive root larger than

α

a negative root

no positive root

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detailed solution

Correct option is A

Let $f (x) = a_{n} x^{n} + a_{n - 1} x^{n - 1} + \dots + a_{1} x$ . It is given that $α$ is a positive root of $f (x) = 0$ .

Also, we observe that $f (0) = 0 .$ Thus, $x = 0$ and $x = α$ are two roots of $f (x)$

By Rolle's theorem, $f^{'} (x)$ has at least one real root between 0 and $α .$

$∴ n a_{n} x^{n - 1} + (n - 1) a_{n - 1} x^{n - 2} + \dots + a_{1} = 0$

has at least one real root between 0 and $α .$

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If the equation anxn+an−1xn−1+…+a1x=0 has a positive root α, then the equation nanxn−1+(n−1)an−1xn−2+…+a1=0 has