Real analysis
We are learning Rolle, Lagrange, Fermat, Taylor Theorems in our Real Analysis class. We just finished continuity and are now studying differentiation. We are using the books by Rudin, Ross, Morrey/Protter.
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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.
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Cx is a constant dependent on x ,
x0 is a particular x in (a,b)
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This is a proof regarding a twice differentiable function.
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