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Let f be a differentiable function from R to R such that `|f(x) - f(y) | le 2|x-y|^(3/2),` for all `x, y in R`. If `f(0) =1,` then `int_(0)^(1) f^(2)(x) dx` is equal to

A

0

B

2

C

`1/2`

D

1

Text Solution

Verified by Experts

The correct Answer is:
D
`(|f(x)-f(y)|)/(|x-y|)le2(x-y)^(1//2)`
`lim_(xrarry)|(f(x)-f(y))/(x-y)|le lim_(xrarry)2|(x-y)|^(1//2)`
`|f'(0)|le0" "rArr" "f'(x)=0`
f is constant
`f(x)=1 AA x`
`int_(0)^(1)(f(x))^(2)=dx=1`
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