InterviewSolution
Saved Bookmarks
InterviewSolution
| 1. |
Statement -1: Let f(x) be a function satisfying `f(x-1)+f(x+1)=sqrt(2)f(x)` for all ` x in R ` . Then f(x) is periodic with period 8. Statement-2: For every natural number n there exists a periodic functions with period n.A. Statement-1 is True, Statement-2 is True, statement-2 is a correct explanation for the statement-1 .B. Statement-1 is True, Statement-2 is True, statement-2 is not a correct explanation for the statement-1 .C. Statement-1 is True, Statement-2 is False.D. Statement-1 is False , Statement-2 is True. |
|
Answer» Correct Answer - B Function f(x) satisfies the relation. `f(X-1)+f(x+1)=sqrt(2)f(x)` for all x. ……..(i) Replacing x by `x+1` in (i), we get . `f(x)+f(x+2)=sqrt(f)(x+1)` ………(ii) Replacing x by (x-1) in (i) ,we get `f(x-2)+f(x) =sqrt(2)f(x-1)` …..(iii) Adding (ii) and (iii) , we get `2f(x)+f(x+2)+f(x-2)=sqrt(2)(f(x-1)+f(x+1))` `implies 2f(x)+f(x+2)+f(x-2)=sqrt(2)xxsqrt(2)f(x)` `implies f(x+2)+f(x-2)=0` Replacing x by x+2 in (ii) we get `f(x+4)+f(x)-0` Replacing x by x+4 in (iv) we get `f(x+8)+f(x+4)=0` Subtracting (iv) from (v), we get `f(x+8)-f(x)=0implies f(x)=f(x+8)` So, f(x) is a periodic function with period 8. Hence, statement-1 is true. We know that g(x)=x-[x] is a periodic function with period 1. Therefore, for any ` n in N, phi (x)=(x)/(n)-[(x)/(n)]` is periodic with period n. |
|