Now, as f(x) is a polynomial so it is continuous and differentiable in R. and as f(0)=0 and
f(1)=a0+
a1
2
+
a3
3
+...+
9n
n+1
=0 (given)
So, according to Rolle's theorem f′(x)=0 for at least one value of x∈(0,1). Therefore, Statement (I) is true. It is given that [f:a,b]→R is continuous on [a,b] and differentiable in (a,b), where a>0 and
f(a)
a
=
f(b)
b
⇒f(b)=
b
a
f(a) Now, according to By Lagrange's mean value theorem (L.M.V.T), we have