Question

# Let f(x) and ϕ(x) are two continuous functions on R satisfying ϕ(x)=x∫af(t) dt,a≠0 and another continuous function g(x) satisfying g(x+α)+g(x)=0 ∀ x∈R,α>0, and 2k∫bg(t) dt is independent of b.  Least positive value of c if c,k,b are in A.P. is01α 2α

Solution

## The correct option is D 2α g(x+α)+g(x)=0⇒g(x+2α)+g(x+α)=0⇒g(x+2α)=g(x) Thus, g(x) is periodic with period 2α. ∴2k∫bg(t) dt=b+c∫bg(x) dx[∵b,k,c are in A.P.] This is independent of b. Then c has least value of 2α.

Suggest corrections