Let f(x) be positive, continuous, and di...

Let `f(x)` be positive, continuous, and differentiable on the interval `(a , b)a n d("lim")_(xveca^+)f(x)=1,("lim")_(xvecb^-)f(x)=3^(1/4)dotIff^(prime)(x)geqf^3(x)+1/(f(x)),` then the greatest value of `b-a` is `pi/(48)` (b) `pi/(36)` `pi/(24)` (d) `pi/(12)`

`(pi)/48`

`(pi)/36`

`(pi)/24`

`(pi)/12`

Text Solution

Verified by Experts

The correct Answer is:

`f'(x)gef^(3)(x)+1/(f(x))`
or `(f(x)f'(x))/(1+f^(4)(x))xge1`
Integrating on the interval `(a,b)` we get
`int_(a)^(b)(f(x)f'(x))/(1+f^(4)(x))dxge int_(a)^(b)dx`
or `[1/2 tan^(-1)f^(2)(x)]_(a)^(b)geb-a`
or `b-a le 1/2[int_(xto b^(-))tan^(-1)f^(2)(x)-lim_(xtoa^(+))tan^(-1) f^(2)(x)]`
`=1/2[tan^(-1)sqrt(3)-tan^(-1)1]`
`=(pi)/24`

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