Let $f(x)$ be a real-valued function that is thrice differentiable on $[a,b]$, prove there exists a $\xi \in (a,b)$ such that $$f(b)=f(a)+\frac{1}{2}(b-a)(f'(a)+f'(b))-\frac{1}{12}(b-a)^3f'''(\xi)$$ I failed to establish a fine auxiliary function and the Taylor formula seems useless here. Please help me with your fabulous auxiliary function :)
Asked
Active
Viewed 107 times
1
-
1You may draw a wrong conclusion.Consider $f(x)=x^2$ on $[a,b]=[0,1]$ – kellty Dec 15 '17 at 02:31
-
If $a$ and $b$ are independent of each other, how are the coefficients of the first derivatives and abscissae just constant numbers? – Reza Dec 15 '17 at 02:31
-
@kellty sorry I've left out something and I just corrected it – Mengfan Ma Dec 15 '17 at 03:22
-
@Reza yes, the original question got something wrong, I just corrected it – Mengfan Ma Dec 15 '17 at 03:23
-
Isn't this trapezoidal rule? – Paramanand Singh Dec 15 '17 at 04:40
-
Your equation should be $f(b) =f(a) +\dots$ instead of $f(a) =f(b) +\dots $. Other terms on right hand side are OK. I have edited to fix this typo. – Paramanand Singh Dec 15 '17 at 04:44