Reference no: EM13128606
Context: We are learning Rolle, Lagrange, Fermat, Taylor Theorems in our Real Analysis class. We just finished continuity and are now studying differentiation.
Question:
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.
Notes:
Cx is a constant dependent on x ,
x0 is a particular x in (a,b)
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