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For every function f (x) which is twice ...

For every function f (x) which is twice differentiable , these will be good approximation of
`int_(a)^(b)f(x)dx=((b-a)/(2)){f(a)+f(b)}`, for more acutare results for ` cin(a,b),F( c) = (c-a)/(2)[f(a)-f( c)]+(b-c)/(2)[f(b)-f( c)]`
When ` c= (a+b)/(2)`
`int_(a)^(b)f(x)dx=(b-a)/(4){f(a)+f (b)+2 f ( c) }dx`
If `lim_(t toa) (int_(a)^(t)f(x)dx-((t-a))/(2){f(t)+f(a)})/((t-a)^(3))=0`, then degree of polynomial function f (x) atmost is

A

0

B

1

C

3

D

2

Text Solution

Verified by Experts

The correct Answer is:
B
Given `underset(t toa)(lim)(int_(a)^(t)f(x)dx -((t-a))/(2){f(t)+f(a)})/((t-a)^(3))=0`
Using L ' Hospital ' s rule , put t - a = h
`rArr underset(h to0)(lim)(int_(a)^(a+h)f(x)dx-(h)/(2){f(a+h)+f(a)})/(h^(3))=0`
`rArrunderset(h to0)(lim)(f(a+h)-(1)/(2){f(a+h)+f(a)}-(h)/(2){f'(a+h)})/(3h^(2))=0`
Again , using L ' Hosipital 's rule,
`underset(h to 0)(lim)(f'(a+h)-(1)/(2)f'(a+h)-(1)/(2)f'(a+h)-(h)/(2)f'' (a+h))/(6h)=0`
`rArrunderset(h to 0)(lim)(-(h)/(2)f''(a+h))/(6 h)=0`
`rArrf''(a)=0,AA a in R`
`rArr` f(x) must have maximum degree 1.
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