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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].If `int_0^x sqrt[1-(f'(t))^2].dt`=`int_0^x f(t).dt`, `0<=x<=1` and f(0)=0,then

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)gt1/3`

C

`f(1/2)lt1/2` and `f(1/3)lt1/3`

D

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

Text Solution

Verified by Experts

The correct Answer is:
C
`f'=+-sqrt(1-f^(2))`
or `f(x)=sinx `or `f'(x)=-sinx` (not possible)
`:. f(x)=sinx`
Also `xgtsinxAAxgt0`
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Knowledge Check

  • Let f be a non-negative function defined on the interval [0,1],ifint_0^xsqrt(1-[f'(t)]^2dt=int_0^xf(t)dt,0lexle1 and f(0)=0, then

    A
    `f(1/2)lt1/2` and `f(1/3)gt1/3`
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    `f(1/2)gt1/2` and `f(1/3)gt1/3`
    C
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    D
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  • Let f be a real valued function defined on the interval [0,infty] by f(x) = logx + int_(0)^(x) sqrt(1+ sin t).dt then which is (are) true:

    A
    `f''(x)` exists `AA x in [0, infty]`
    B
    `f'(x)` exists `AA x in [0,pi]` and f' is continous on `[0, infty]` but not differentiable on `[0, infty]`
    C
    `alpha gt 1` such that `|f'(x)| lt |f(x)| AA x in [alpha, infty|`
    D
    `3 beta gt 0` such that `|f(x)| + |f'(x)| le beta AA x in [0, infty]`

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    D
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