KVPY Sample Paper KVPY Stream-SX Model Paper-8

If \[f=R\to R\] and for a fixed positive number c ; \[f(x+c)=1+[1-5f(x)+10{{\{f(x)\}}^{2}}\]\[-10\{{{(f(x)\}}^{3}}+5{{\{f(x)\}}^{4}}-\{f(x)\}5]\,\,1/5\,\,\forall \,\,\]\[x\in R,\] then f(x) is a periodic function whose period can be -

A) c

B) 2c

C) 3c

D) 5c

Correct Answer: B

Solution :

Given \[f\,\,(x+c)=1+{{[{{(1-f\,\,(x))}^{5}}]}^{1/5}}\]

\[\Rightarrow f\,\,(x+c)=1+[\,1-f\,\,(x)]\]

\[\Rightarrow f\,\,(x+2c)=1+[1-f\,\,(x+c)]\]

\[=1+[\,1-1-1+f\,\,(x)]\]

\[=1+f\,\,(x)-1=f\,\,(\,x+2c)=f\,\,(x)\]

Hence f(x) is a periodic function of period 2c.