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Question

Let f(x) and ϕ(x) are two continuous functions on R satisfying ϕ(x)=xaf(t) dt,a0 and another continuous function g(x) satisfying g(x+α)+g(x)=0  xR,α>0, and 2kbg(t) dt is independent of b

Least positive value of c if c,k,b are in A.P. is
  1. 0
  2. 1
  3. α 
  4. 2α 


Solution

The correct option is D 2α 
g(x+α)+g(x)=0g(x+2α)+g(x+α)=0g(x+2α)=g(x)

Thus, g(x) is periodic with period 2α.
2kbg(t) dt=b+cbg(x) dx[b,k,c are in A.P.]

This is independent of b. Then c has least value of 2α.

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