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If `f: Rvec[0,oo)i safu n c t ions u c ht h a tf(x-1)+f(x+1)=sqrt(3)f(x),`then prove that `f(x)`is periodic and find its period. |
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Answer» `f(x-1) +f(x+1)=sqrt(3) f(x) " (1)" ` Putting `x+2` for x in relation (1), we get `f(x+1) +f(x+3)=sqrt(3) f(x+2) " (1)" ` From (1) and (2), we get `f(x-1) +2f(x+1)+f(x+3) =sqrt(3)(f(x)+f(x+2))` `=sqrt(3)(sqrt(3)f(x+1))` `=3f(x+1)` `or f(x-1)+f(x+3)=f(x+1) " (3)" ` Putting `x+2` for x in (3), we get `f(x+1)+f(x+5)=f(x+3) " (4)" ` Adding (3) and (4), we get `f(x-1)= -f(x+5).` Now, put `x+1` for x. Then `f(x)= -f(x+6) " (5)" ` Put `x+6` in place of x in (5). Then `f(x+6)= -f(x+12).` Therefore, from (5) again, `f(x)= -[-f(x+12)]=f(x+12)`. Hence, the period of `f(x)` is 12. |
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