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.
******************************************************
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.
****************************************************
Cx is a constant dependent on x ,
x0 is a particular x in (a,b)
******************************************************
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.
- Plain text response
- Attached file(s):
- 2879.doc
- Your Information is Secure
- Best Online Academic Help Service
- Students find real academic Success
- Real Analysis : Proof - I need a proof for "If f on [a,b] is continuous & 0 is not a member f([a,b]) then f is bounded away from 0."
- real analysis - taking the continuity of h(x) as given in#30026,#30028 by using any of the functional limits and continuity theorems prove that the finite sum g_m (x)=sum sign(oo top n=0 bottom) of 1/2^n h(2^n x) ...
- Continuity Proof Relating to Cantor Set - The attached file explains. I would just like a simple proof for continuity, if indeed the function is continuous.
- Real Analysis Problem - We have learned Rolle, Lagrange, Fermat, Taylor Theorems in our Real Analysis class and we have finished differentiation. We just started integration. In this problem we are not supposed to use any ma ...
- Real Analysis (Uniform Continuity) - Explain the given proof all S(1) ~ S(5). Please explain clearly and specifically. Proof. S(1): Let ε=1, and let any δ>0 be given. S(2): Let n be an integer > max(1, 1/δ), and s ...
