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If f is a real- valued differentiable fu...

If f is a real- valued differentiable function satisfying `|f(x) - f(y)| le (x-y)^(2) ,x ,y , in R and f(0) =0` then f(1) equals

A

`-1`

B

0

C

2

D

1

Text Solution

Verified by Experts

The correct Answer is:
B
`|f(x)-f(y)|lt=(x-y)^(2)`
`implies |f(x+h)-f(x)|lt=h^(2)implies (|f(x+h)-f(x)|)/(|h|)lt=h`
(Dividing by ‘h’ (positive real No.)
`lim_(hrarr0)|(f(x+h)-f(x))/(h)|lt=0`
`implies |f(x) lt=0 implies f(x) =0 implies f(x) =c ` (Where c is only constant)
f(0) =0 Thus c =0 f(1) =0
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