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Let f be a non-negative function defined...

Let `f` be a non-negative function defined on the interval `[0,1]dot` If `int_0^xsqrt(1-(f^(prime)(t))^2)dt=int_0^xf(t)dt ,0lt=xlt=1,a n df(0)=0,t h e n` `f(1/2)<<1/2a n df(1/3)>>1/3` `f(1/2)>1/2a n df(1/3)>1/3` `f(1/2)<1/2a n df(1/3)<1/3` `f(1/2)>1/2a n df(1/3)<1/3`

A

`f((1)/(2))lt(1)/(2)and f((1)/(3))gt (1)/(3)`

B

`f((1)/(2))gt(1)/(2)and f((1)/(3))gt (1)/(3)`

C

`f((1)/(2))lt(1)/(2)and f((1)/(3))lt (1)/(3)`

D

`f((1)/(2))gt(1)/(2)and f((1)/(3))lt(1)/(3)`

Text Solution

Verified by Experts

The correct Answer is:
C
Given `int_(0)^(x)sqrt(1-{f'(t)}^(2))dt = int_(0)^(x)f (t) dt , 0 le x le 1`
Differentiating both sides W. r . T. X by using Leidnitz 's rule , we get
`sqrt(1-{f'(x)}^(2))= f (x) rArr f ' (x) = +- sqrt(1-(f(x)}^(2))`
`rArr int (f' (x))/(sqrt(1-{f(x)}^(2)))dx =+-int dx rArr sin^(-1){ f(x)}=+- x+c`
Put `x=0rArrsin^(-1){f(0)}=c`
` rArr c= sin^(-1)(0) =0`
[`:' f(0)=0]`
`:. f(x) =+- sinx`
But `f(x) ge0,AAx in[0,1]`
` :. f(x) = sin x`
As we know that ,
`sinx ltx, AA x lt 0`
`:. sin((1)/(2))lt(1)/(2) andsin ((1)/(2))lt(1)/(3)`
` rArr f((1)/(2))lt(1)/(2)andf((1)/(3))lt(1)/(3)`
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