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If the equation a(n) x^(n) + a(n -1) x^(...

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-1) + ..... , a_(1) = 0 ` has a positive root, which is :

A

greater than or equal to `alpha`

B

equal to `alpha`

C

greater than `alpha`

D

smaller than `alpha`

Text Solution

Verified by Experts

The correct Answer is:
D
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`.
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