given-f-R-R-f-x-a-n-x-n-a-0-with-a-n-0-a-i-R-i-0-1-n-n-N-supose-that-x-1-x-n-are-roots-the-n-roots-of-f-x-then-proof-or-give-a-counter-example-that-the-root-of-f-n-1-x-

Question Number 1120 by 123456 last updated on 17/Jun/15

given f : R \to R

f (x) = a_{n} x^{n} + \cdot \cdot \cdot + a_{0}

with a_{n} \neq 0, a_{i} \in R, i \in {0, 1, \dots, n}, n \in N^{*}

supose that x_{1}, \dots, x_{n}

are roots the n roots of f (x)

then proof or give a counter example that the root of

f^{(n - 1)} (x)

is

x = \frac{x_{1} + \cdot \cdot \cdot + x_{n}}{n}

Answered by prakash jain last updated on 17/Jun/15

f (x) = (x - x_{1}) (x - x_{2}) \dots (x - x_{n})

f (x) = x^{n} - (x_{1} + x_{2} + . . + x_{n}) x^{n - 1} + \dots + x_{1} x_{2} . . x_{n}

f^{n - 1} (x) = n (n - 1) (n - 2) \dots 2 x - (n - 1)! (x_{1} + x_{2} + \dots + x_{n})

Root of f^{n - 1} (x) = \frac{x_{1} + x_{2} + \dots + x_{n}}{n}