IIT JEE Advanced 2006 Paper 1

For every function f(x) which is twice differentiable, these will be good approximation of

b

∫

a

f(x)dx≅(‌

b−a

2

){f(a)+f(b)}.
Now, if we take c=‌

a+b

2

then using the above again, we get

b

∫

a

f(x)dx=

c

∫

a

f(x)dx+

b

∫

c

f(x)dx≅‌

b−a

4

{f(a)+f(b)+2f(c)}
and so on.
We get approximation for value of

b

∫

a

f(x)dx.
If f ′′(x) < 0, ∀x ∈(a, b), c(c, f (c)) is point of maxima, where c ∈ (a, b), then f ′(c) is