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
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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