Let f be a real valued function with domain R satisfying f (x + k) = 1 + [2 – 5 f (x) + 10 { f (x)}²– 10 { f (x)}³ + 5 { f (x)}⁴ – { f (x)}5]^1/5
for all real x and some positive constant k, then the period of the function f (x) is

Question

Infinity Learn · Accepted Answer

We have,f (x$$ $$+$$ $$k) $$=$$ $$1$$ $$+$$ [$$1$$ $$+$$ $${1$$ – f (x)}5$$]$$1/5$$⇒ f $$(x$$ $$+$$ $$k) – $$1$$ $$=$$ [$$1$$ – $$($$ f (x) – $$1)5$$]$$1/5$$⇒ϕ(x+k)=1$$−$${$$ϕ$$(x)}51/5$$ where  ϕ$$(x)=f(x)−$$1$$⇒ϕ(x+2k)=1$$−$${$$ϕ$$(x+k)}51/5$$⇒ϕ$$(x+2k)=1$$−$$1$$−$$($$ϕ$$(x))51/5=$$ϕ$$(x),∀x∈R⇒ f (x$$ $$+$$ $$2k) – $$1$$ $$=$$ f (x) – $$1$$⇒ f (x$$ $$+$$ $$2k) $$=$$ f (x), ∀ x ∈ R∴ f (x) is periodic with period $$2k$$.

Let f be a real valued function with domain R satisfying f (x + k) = 1 + [2 – 5 f (x) + 10 { f (x)}²– 10 { f (x)}³ + 5 { f (x)}⁴ – { f (x)}5]^1/5
for all real x and some positive constant k, then the period of the function f (x) is

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Let f be a real valued function with domain R satisfying f (x + k) = 1 + [2 – 5 f (x) + 10 { f (x)}2 – 10 { f (x)}3 + 5 { f (x)}4 – { f (x)}5]1/5for all real x and some positive constant k, then the period of the function f (x) is

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Let f be a real valued function with domain R satisfying f (x + k) = 1 + [2 – 5 f (x) + 10 { f (x)}²– 10 { f (x)}³ + 5 { f (x)}⁴ – { f (x)}5]^1/5
for all real x and some positive constant k, then the period of the function f (x) is