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If f(x) satisfies the requirements of La...

If f(x) satisfies the requirements of Lagrange's mean value theorem on [0, 2] and if f(0)= 0 and `f'(x)lt=1/2`

A

`f(x) le 2`

B

`|f (x)| le 1`

C

`f(x) =2x`

D

`f(x)="3 for at least one x in [0,2]"`

Text Solution

Verified by Experts

The correct Answer is:
B
`(f(2)-f_(0))/(2-0)=f' (x) rArr (f(2)-0)/(2)=f'(x)`
`rArr (df(x))/(dx)=(f(2))/(2) rArr f(x)=(f(2))/(2)x+c`
`therefore f(0)=0 rArr c=0, therefore f(x)=(f(2))/(2)x ......(i)`
Given `|f'(x) le 1/2 rArr |(f(2))/(2)| le 1/2 ........(ii)`
(i) `|f(x)=|(f(2))/(2)x|=|(f(2))/(2)| |x| le 1/2 |x| "(from (ii))"`
In [0,2] for maximum `x(x=2) rArr |f(x)| le1`
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