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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

2

C

3

D

does not exist

Text Solution

Verified by Experts

The correct Answer is:
C
`f(1+h)=f(1)+hf'(1)+(1)/(2)h^(2)f''(c)`
`therefore" "f(1+h)=1+3h^(2)c`
`therefore" "(f(1+h)-f(1))/(h^(2))=3e`
`therefore" "lambda=3`
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