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Let f(x) and phi(x) are two continuous f...

Let `f(x)` and `phi(x)` are two continuous function on `R` satisfying `phi(x)=int_(a)^(x)f(t)dt, a!=0` and another continuous function `g(x)` satisfying `g(x+alpha)+g(x)=0AA x epsilonR, alpha gt0`, and `int_(b)^(2k)g(t)dt` is independent of `b`
Least positive value fo `c` if `c,k,b` are n A.P. is

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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CENGAGE-DEFINITE INTEGRATION -Exercise (Comprehension)
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