# 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

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...ed).

So if we show that there exist c_x such that H''(c_x)=0, this means:

g(x)f''(c_x)+f(x)=0 i.e. f(x)=-g(x)f''(c_x) which is what we need to show.

So all we have to show is that there exists such a point in which H'' equals ...