Answer to Question #314397 in Real Analysis for Pankaj

Question

Let f: [0, 1]→R be a function defined by f(x) = x^m (1-x)^n ,where . m, n∈N

Find the values of m and n such that the Rolle’s Theorem holds for the function

f .

Answer 1

The function is differentiable on $\left[ 0,1 \right]$ , since it is a polynomial.

$f\left( 0 \right) =0,f\left( 1 \right) =0$ .

Hence the Rolle’s theorem holds for any $m,n\in \mathbb{N}$