if f''(x) exist on [a,b] and f(a)=f(b) . prove that f((a+b)/2)=(f(a)+f(b))/2 +((b-a)^2)*f''(c). for c belongs to (a,b).

Question

anonymous · Answer

without writing it i will bet that it is mvt (or rolle's theorem) twice yes?

jamesj · Answer

Are you sure you have written this expression down correctly?

For example the term (f(a)+f(b))/2 = f(a) = f(b) ?

anonymous · Answer

yes

anonymous · Answer

plz help...

zarkon · Answer

Let $$f(x)=x^2$$$$a=-1,b=1$$ then $$f(a)=f(-1)=1=f(1)=f(b)$$

$$f\left(\frac{a+b}{2}\right)=f\left(\frac{-1+1}{2}\right)=f(0)=0^2=0$$

$$\frac{f(a)+f(b)}{2}=\frac{1+1}{2}=1$$

$$f''(c)=2\text{ for all c}$$

then using your formula we get 

$$0=1+(1-(-1))^2\cdot 2\Rightarrow0=9$$

anonymous · Answer

but its a particular case....i need a general ans..

zarkon · Answer

it is a counter example

jamesj · Answer

Pearls my friend, pearls.

jamesj · Answer

@jyoti: What Zarkon is saying is that the relation you have written down cannot be true in general because he has just given us an example of where it is false.

So either you have written down the question incorrectly here in open study; or, the question as given to you is simply incorrect.

anonymous · Answer

may be the question is incorrect....bcz i have written exactly what was given.....