Real analysis

Based on the Rolle, Lagrange, Fermat and Taylor Theorems. ******************************************************
Let f: [a,b] --> R, a < b, twice differentiable with the second derivative continuous such that f(a)=f(b)=0.

Denote M = sup |f "(x)| where x is in [a,b]

and g:[a,b] --> R, g(x)=(1/2)(x-a)(b-x)


i) Prove that for all x in [a,b], there exists

Cx in (a,b) such that f(x)= - f "(Cx)g(x).

Cx is a constant dependent on x