(i) Calculate the Fourier series for the periodic extension on $\mathbb{R}$ of the function

$$f(x)=x(1-x)$$

defined on the interval $[0,1)$.

(ii) Explain why the Fourier series for the periodic extension of $f^{\prime}(x)$ can be obtained by term-by-term differentiation of the series for $f(x)$.

(iii) Let $G(x)$ be the Fourier series for the periodic extension of $f^{\prime}(x)$. Determine the value of $G(0)$ and explain briefly how it is related to the values of $f^{\prime}$.