question_answer

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 [AIEEE 2005]

A) Greater than or equal to a

B) Equal to \[\alpha \]

C) Greater than \[\alpha \]

D) Smaller than \[\alpha \]

Correct Answer: D

Solution :

Let \[f(x)={{a}_{n}}{{x}^{n}}+{{a}_{n-1}}{{x}^{n-1}}+....+{{a}_{1}}x\];      \[f(0)=0;\] \[f(\alpha )=0\] Þ \[{f}'(x)=0\], has atleast one root between \[(0,\alpha )\] i.e.,  equation \[n{{a}_{n}}{{x}^{n-1}}+(n-1){{a}_{n-1}}{{x}^{n-2}}+....+{{a}_{1}}=0\] has a positive root smaller than .