Question

Let f :[0,1] tends to R be a function defined by f(x)=x^m (1-x)^n, where m,n belongs to N.Find the values of m and n such that the Rolle’s Theorem holds for the function f .

Answer 1

$\forall n,m \in N,f(x)$- Polynomial$\implies$ $f(x)$ -  continuous and differentiable on $(-\infty,\infty)$ $\implies f(x)$ is continuous on the closed interval [0, 1] and differentiable on the open interval (0, 1), $f(0)=0^m *1^n=0,f(1)=1^m*0^n=0,f(0)=f(1)$ $\implies$ the Rolle’s Theorem holds for the function $f(x)$ Answer:$\forall n,m \in N$