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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` and `f(1)=1,` then

A

`f(1) le 0 `

B

`0 le f' (1) le 1/2`

C

`1/2 lt f'(1) le 1`

D

`f(1) gt 1`

Text Solution

Verified by Experts

The correct Answer is:
D
It is given that f is twice differentiable so it is everywherer continuous and differentiable
Consider the function h(x) defined on [0,1]
Clearly h(x) is continuous on [0,1] and differentiable on (0,1) Now `(1/2)=f(1/2)-1/2=1/2-1/2=0` and h(1) =f(1)-1 =1-1 =0
`therefore (1/2)=h(1)` `ltbRgt `Thus h(x) satisfies the conditions of Rolles theorem Consequently there exists `alpha in R`
`rArr f(x)` is increasing on R
`rArr f(1) gt f(alpha)`
`rArr f(1) gt 1.`
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