InterviewSolution
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InterviewSolution
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If `a , b`are two fixed positive integers such that `f(a+x)=b+[b^3+1-3b^2f(x)+3b{f(x)}^2-{f(x)}^3]^(1/3)`for all real `x ,`then prove that `f(x)`is periodic and find its period. |
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Answer» `f(a+x)=b+[b^(3)+1-3b^(2)f(x)+3b{f(x)}^(2)-{f(x)}^(3)]^(1//3)` `=b+[1+{b-f(x)}^(3)]^(1//3)` `or f(a+x)-b=[1-{b-f(x)}^(3)]^(1//3)` ` or phi(a+x)=[1-{phi(x)}^(3)]^(1//3) " (1)" ` where `phi (x)=f(x)-b` `or phi (2a+x)=[1-{phi(x+a)}^(3)]^(1//3)=phi(x) " From (1)]" ` `or f(x+2a)-b=f(x)-b` `or f(x+2a)=f(x)` Therefore, `f(x)` is periodic with period 2a. |
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