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Let f be a continuous and differentiable...

Let f be a continuous and differentiable function in `(x_(1),x_(2))`. If `f(x).f'(x)ge x sqrt(1-(f(x))^(4))` and `lim_(xrarrx_(1))(f(x))^(2)=1 and lim_(xrarrx) )(f(x))^(2)=(1)/(2)`, then minimum value of `(x_(1)^(2)-x_(2)^(2))` is

A

`(pi)/(6)`

B

`(2pi)/(3)`

C

`(pi)/(3)`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
C
`(2f(x).f'(x))/(sqrt(1-(f(x))^(4)))-2xge0`
`rArr" "(d)/(dx)(sin^(-1)(f(x))^(2)-x^(2))ge0`
Then `g(x)=sin^(-1)((f(x))^(2))-x^(2)` is a non-decreasing function.
`rArr" "underset(xrarrx_(1)^(+))(lim)g(x)le underset(xrarrx_(2)^(-))(lim)g(x)`
`rArr" "(pi)/(2)-x_(1)^(2)le(pi)/(6_-x_(2)^(2)`
`rArr" "x_(1)^(2)-x_(2)^(2)ge(pi)/(3)`
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