Find the period of f(x) satisfying the condition :
(i) f(x + p) = 1 + {1 – 3 f(x) + 3 f 2 (x) – f 3 (x)} 1/3 , p > 0
(ii)f(x – 1) + f(x + 3) = f(x + 1) + f(x + 5)
Text Solution
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(i) 2p (ii) 8
Sol. (i) f(x + p) = {1 + (1 – f(x)) 3 } 1/3
⇒ f(x + p) + f(x) = 2 .......(i)
replacing x by (x + p) in equation (i)
f(x + 2p) + f(x + p) = 2 ......(ii)
from (i) and (ii)
f(x) = f(x + 2p) ⇒ period = 2p
(ii) f(x – 1) + f(x + 3) = f(x + 1) + f(x + 5) .....(i)
Replace x by x + 2
f(x + 1) + f(x + 5) = f(x + 3) + f(x + 7) ......(ii)
(i) – (ii)
f(x – 1) = f(x + 7) ⇒ f (x + 8) = f(x)
period = 8
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