Let f be a real -valued function with domain R satisfying
f(x + 2008) = 1 +3√ 2 - 3f(x) + 3f(x)2 - f(x)3
Answer the following
1) The period of f(x) is
a) 2008 b)4016 c) 1004 d) 0
2) The value of f(x+4016) is
a) 2008 b) 4016 c)1004 d)0
3) The value of [f(0) - f(4016)] + [f(4016) - f(8032)] + [f(8032) - f(12048)] +..........+2011 times is equal to
a) 2008x2011 b) 4016x2011 c) 2009x2011 d)0
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2 Answers
62\\f(x+2008)=1+{[1-(f(x)-1)^3]}^{1/3} \\(1-f(x+2008))^3={(f(x)-1)^3-1} \\(1-f(x))^3={(f(x-2008)-1)^3-1} \text{...replacing x by x-2008}
hence using the above two, we get that f(x-2008)=f(x+2008)
so the period is 4016