InterviewSolution
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InterviewSolution
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Let f be a real valued periodic function defined for all real umbers x such that for some fixed ` a gt 0` , `f(x+a)=(1)/(2)+sqrt(f(x)-{f(x)}^(2))` for all x . Then , the period of f(x) isA. aB. 2aC. 3aD. 4a |
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Answer» Correct Answer - B We have , `f(x+a)=(1)/(2)+sqrt(f(x)-{f(x)}^(2))` `:. f(x+a+a)=(1)/(2)+sqrt(f(x+a)-{f(x+a)}^(2))` `implies f(x+2a)=(1)/(2)+sqrt(f(x+a){1-f(x+a)})` `impliesf(x+2a)=(1)/(2)+ sqrt({(1)/(2)+sqrt(f(x)-{f(x)}^(2))}{(1)/(2)-sqrt(f(x)-{f(x)}^(2))})` `implies f(x+2a)+(1)/(2)+sqrt((1)/(4)-{f(x)-{f(x)}^(2)})` `implies f(x+2a)=(1)/(2)+sqrt((1)/(4)-f(x)+{f(x)}^(2))` `implies f(x+2a)=(1)/(2)+sqrt({f(x)-(1)/(2)}^(2))` `implies f(x+2a)=(1)/(2)+|f(x)-(1)/(2)|" "[ :. sqrt(x^(2))=|x|]` `implies f(x+2a)=(1)/(2)+f(x)-(1)/(2)" " [ :. f(x) ge (1)/(2)]` `implies f(x+2a)=f(x)` for all x . Hence, f(x) is a periodic function with period 2a. |
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