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If f(x+f(y))=f(x)+y for all x,y in R and...

If f(x+f(y))=f(x)+y for all `x,y in R` and f(0)=1, then `int_(0)^(10) f(10-x)dx` is equal to

A

1

B

10

C

`underset(0)overset(1)int f(x) dx`

D

`10 underset(0)overset(1)int f(x)dx`

Text Solution

Verified by Experts

The correct Answer is:
D
We have,
f(x+f(y))=f(x)+y for all `x,y in R`
Replacing y by 0, we obtain
`rArr f(x+f(0))=f(x)+0` for all ` xin R`
`rArr f(x+1)=f(x)` for all `x in R`
`rArr f(x)` is periodic with period 1.
Now,
`underset(0)overset(10)int f(10-x)dx=-underset(10)overset(0)int f(t) dt`, where t=10-x
`=underset(0)overset(10)int f(t) dt=10underset(0)overset(1)int f(t)dt[:' f(x)" is periodic with period "1]`
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