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).
ii) Prove that if there exists x0 in (a,b) such that
|f(x0)| = Mg(x0), then f = Mg or f=-Mg.
Cx is a constant dependent on x ,
x0 is a particular x in (a,b)
The solution is comprised of an attachment of two pages of carefully formatted and worked calculations to give the required proof.