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Question

# Let f(x+p)=1+{2−3f(x)+3(f(x))2−(f(x))3}1/3,∀x∈R, where p > 0, then

A
f(x) is periodic with period p
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B
f(x) is periodic with period 2p
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C
f(x) is periodic with period 4p
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D
f(x) is not periodic
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Solution

## The correct option is B f(x) is periodic with period 2pSince, f(x+p)=1+{2−3f(x)+3(f(x))2−(f(x))3}13⟹f(x+p)=1+{1+(1−f(x))3}13⟹f(x+p)−1={1+(1−f(x))3}13⟹f(x+p)−1={1−(f(x)−1)3}13Now, take f(x+p)−1=g(x), we haveg(x+p)=(1−(g(x))3)13g(x+2p)=(1−(g(x+p))3)13⟹g(x+2p)=[1−1+(g(x))3]13⟹g(x+2p)=g(x)⟹f(x+2p)−1=f(x)−1⟹f(x+2p)=f(x).

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