AIEEE 2005 (Offline)Question 1
If the equation $a_nx^{n}+a_{n-1}x^{n-1}+\cdots+a_1x=0$, $a_1\ne0$, $n\ge2$, 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
Differentiation and its Application
Solution
Concept. The second equation is $f'(x)=0$ for $f(x)=a_nx^n+\cdots+a_1x$. Formulas used. (1) Rolle: $f(0)=f(\alpha)\Rightarrow f'(c)=0$ for some $c\in(0,\alpha)$Step 1 - Equal values. $f(0)=0$ and $f(\alpha)=0$. Step 2 - Apply (1). $f'$ has a root $c$ with $0<c<\alpha$: a positive root smaller than $\alpha$. Answer: (B)