Question

let f:[0,1]->R be a function defined by f(x)= x^m (1-x)^n, where m,n belong to N. Find the values of m and n such that the Rolle's theorem holds for the function f.

Answer 1

Given the function $f:[0,1]\to R$

Such that $f(x)= x^m (1-x)^n \space\space m,n \epsilon N$

Since $x^m , f(1-x)^n$ both are continuos on [0,1]

So, f is also continuos on [0,1] and also differentiable of (0,1) because f is a polynomial function and f(0)=f(1)=0

So, we can apply Rolle's theorem

By Rolle's theorem $\varXi c \epsilon (0,1)$ such that f'(c)=0 or $m.c^{m-1}(1-c)^n-nc^m(1-c)^{n-1}=0$

$m.c^{m-1}(1-c)^n=nc^m(1-c)^{n-1}$

$m.c^{m-1-m}=n(1-c)^{n-1-n}$

$\frac{m}{c}=\frac{n}{1-c} \implies m-mc=nc$

$(m+n)c=m$

$c=\frac{m}{m+n} <1$