Let f:R$\to$ R be a periodic function such that $\mathrm{f}(\mathrm{T}+\mathrm{x})=1+{\left[1-3\mathrm{f}\left(\mathrm{x}\right)+3\left(\mathrm{f}\right(\mathrm{x}){)}^2-(\mathrm{f}\left(\mathrm{x}\right){)}^3\right]}^{1/3}$ where T is a fixed positive number, if period of f(x) is kT, then the value of k is ..

Given, $\mathrm{f}(\mathrm{T}+\mathrm{x})=1+{\left[(1-\mathrm{f}(\mathrm{x}){)}^3\right]}^{1/3}=1+(1-\mathrm{f}(\mathrm{x}\left)\right)$

$\begin{array}{l}\mathrm{f}(\mathrm{T}+\mathrm{x})+\mathrm{f}\left(\mathrm{x}\right)=2---\left(\mathrm{i}\right)\\ \mathrm{f}(2\mathrm{T}+\mathrm{x})+\mathrm{f}(\mathrm{T}+\mathrm{x})=2----\left(\mathrm{ii}\right)\\ \left(2\right)-\left(1\right)=\mathrm{f}(2\mathrm{T}+\mathrm{x})-\mathrm{f}\left(\mathrm{x}\right)=0\\ \mathrm{f}(2\mathrm{T}+\mathrm{x})=\mathrm{f}\left(\mathrm{x}\right)\end{array}$

Also, T is positive and least ,therefore period of f(x)=2T
k =2