InterviewSolution
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InterviewSolution
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Let f(x) be a real valued periodic function with domain R such that `f(x+p)=1+[2-3f(x)+3(f(x))^(2)-(f(x))^(3)]^(1//3)` hold good for all ` x in R ` and some positive constant p, then the periodic of f(x) isA. pB. 3pC. 2pD. `p^(2)` |
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Answer» Correct Answer - C We have , `f(x+p)=1+[2-3f(x)+3(f(x))^(2)-(f(x))^(3)]^(1//3)` `implies f(x+p)=1+[1+{1-f(x)}^(3)]^(1//3)` `implies f(x+p)-1=[1-{f(X)-1}^(3)]^(1//3)` ` implies g(x+p)=[1-{g(x)}^(3)]^(1/3)" "`…..(i) where g(x) =f(x)-1. `implies g(x+2p)=[1-{g(x+p)}^(3)]^(1//3)" "`[ On replacing x by x+p] `implies g(x+2p)=[1-{1-g(x)}^(3)]^(1//3)" "`[Using (i)] `implies g(x+2p)=g(x)` for all ` x in R ` `implies g(x+2p)-1=f(x)-1` `implies f(x+2p)=f(x)` Hence , f(x) is a period function with period 2p. |
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