Multiple choice

If the equation an xn + an-1 xn-1 + .......... + a1x = 0, a1 ¹ 0, n ³2 has a positive root x = a, then the equation nanxn-1 + (n-1)an-1xn-2 + ........ + a1 = 0 has a positive root

  1. equal to a

  2. greater than or equal to a

  3. smaller than a

  4. greater than a
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let f(x) = a_n*x^n + ... + a_1*x. f(a) = 0. By Rolle's Theorem, f'(x) = n*a_n*x^(n-1) + ... + a_1 has a root between 0 and a. Thus, the root is smaller than a.

AI explanation

Let f(x) = an xn + an-1 xn-1 + ......... + a1x, where f(0) = 0 and f(a) = 0 for some positive root a. By Rolle's Theorem, if a function is continuous and differentiable on an interval and has equal values at the endpoints, its derivative must be zero somewhere in between. The derivative of the polynomial is f'(x) = nanxn-1 + (n-1)an-1xn-2 + ........ + a1, so there must be a root of f'(x) = 0 in the interval (0, a). Therefore, the derivative equation has a positive root smaller than a.

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