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Let f(x) be a function such that its der...

Let f(x) be a function such that its derovative f'(x) is continuous in [a, b] and differentiable in (a, b). Consider a function `phi(x)=f(b)-f(x)-(b-x)f'(x)-(b-x)^(2)`A. If Rolle's theorem is applicable to `phi(x)` on, [a,b], answer following questions.
If there exists some unmber c(a lt c lt b) such that `phi'(c)=0 and f(b)=f(a)+(b-a)f'(a)+lambda(b-a)^(2)f''(c)`, then `lambda` is

A

`(1)/(2)`

B

`-(1)/(2)`

C

`(1)/(4)`

D

`(1)/(3)`

Text Solution

Verified by Experts

The correct Answer is:
B
`sin(alpha+h)=sin alpha+h cos alpha+(1)/(2)h^(2)(-sin (alpha+th))`
`therefore" "(sin (alpha+h)-sin alpha-h cos alpha)/(h^(2))=-(1)/(2)sin(alpha+th)`
`therefore" "alpha=-(1)/(2)`
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