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Real analysis

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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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Solution Summary

This is a proof regarding a twice differentiable function.

Solution Preview

Let x be a fixed point in the interval [a,b] and consider the function of y:
H(y)=g(x)f(y)-g(y)f(x). Then notice that H'(y)=g(x)f'(y)-g'(y)f(x) and H''(y)=g(x)f''(y)-g''(y)f(x)=g(x)f''(y)+f(x) (because g''=-1 as can easily be ...

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