If the equation `a_(n)x^(n)+a_(n-1)x^(n-1)+..+a_(1)x=0, a_(1)!=0, n ge2`, has a positive root `x=alpha` then the equation `na_(n)x^(n-1)+(n-1)a_(n-1)x^(n-2)+….+a_(1)=0` has a positive root which is

Options

A. greater than or equal to `alpha`
B. equal to `alpha`
C. greater than `alpha`
D. smaller than `alpha`

Answer

D

Explanation

Let `f(x)=a_(n)x^(n)+a_(n-1)x^(n-1)+……….+a_(1)x`
`f(0)=0,f(alpha)=0`
`f'(x)=0` has atleast one root between `(0,alpha)`
i.e. Equation
`na_(n)x^(n-1)+(n-1)a_(n-1)x^(n-2)+….+a_(1)=0`
has a positive root smaller than `alpha`.