Algebra of real valued functions
Question

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

Difficult
Solution

## Given, $\mathrm{f}\left(\mathrm{T}+\mathrm{x}\right)=1+{\left[\left(1-\mathrm{f}\left(\mathrm{x}\right){\right)}^{3}\right]}^{1/3}=1+\left(1-\mathrm{f}\left(\mathrm{x}\right)\right)$$\begin{array}{l}\mathrm{f}\left(\mathrm{T}+\mathrm{x}\right)+\mathrm{f}\left(\mathrm{x}\right)=2---\left(\mathrm{i}\right)\\ \mathrm{f}\left(2\mathrm{T}+\mathrm{x}\right)+\mathrm{f}\left(\mathrm{T}+\mathrm{x}\right)=2----\left(\mathrm{ii}\right)\\ \left(2\right)-\left(1\right)=\mathrm{f}\left(2\mathrm{T}+\mathrm{x}\right)-\mathrm{f}\left(\mathrm{x}\right)=0\\ \mathrm{f}\left(2\mathrm{T}+\mathrm{x}\right)=\mathrm{f}\left(\mathrm{x}\right)\end{array}$Also, T is positive and least ,therefore period of f(x)=2Tk =2

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