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If f(x) is a function satisfying f(x+a)+...

If `f(x)` is a function satisfying `f(x+a)+f(x)=0` for all `x in R` and positive constant `a` such that `int_b^(c+b)f(x)dx` is independent of `b ,` then find the least positive value of `cdot`

A

`0`

B

`1`

C

`alpha`

D

`2alpha`

Text Solution

Verified by Experts

The correct Answer is:
D
`g(x+alpha)+g(x)=0`
or `g(x+2alpha)+g(x+alpha)=0`
or `g(x+2alpha)=g(x)`
Thus `g(x)` is periodic with period `2alpha`.
`:. int_(b)^(2k)g(t)dt=int_(b)^(b+c)g(x)dx` ( `:'b,k,c` in A.P.)
This is independent of b. Then c has least value `2alpha`.
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  14. If f(x)=int0^1(dt)/(1+|x-t|),t h e nf^(prime)(1/2)i se q u a lto 0 (b...

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  15. If f(x)=int(0)^(1)(dt)/(1+|x-t|),x epsilonR Which of the following ...

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  16. Let f be a differentiable function satisfying int(0)^(f(x))f^(-1)(t)d...

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  17. Let f be a differentiable function satisfying int(0)^(f(x))f^(-1)(t)d...

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  19. If Un=int0^pi(1-cosnx)/(1-cosx)dx , where n is positive integer or zer...

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  20. Evaluate int(0)^(2)(2x^2+x+1)dx

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