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If f:R->R is a twice differentiable func...

If `f:R->R` is a twice differentiable function such that `f''(x) > 0` for all `x in R, and f(1/2)=1/2. f(1)=1,` then

A

`0ltf'(1)le1/2`

B

`f'(1)le0`

C

`f'(1)>1`

D

`1/2ltf'(1)le1`

Text Solution

Verified by Experts

The correct Answer is:
3
using LMVT on f(x) for `x in [1/2,1]`
`f(1)-f(1/2)/(1-1/2)=f(c )` where `c in (1/2,1)`
`rarr 1-(1)/(2/(1/2))=f(c )`
`rarr` f(c )=1 where c `in (1/2,1)`
Now `f(x)gt0` so f(x) is increasing function forall x in R`
`therefore f(1)gt1`
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