Define $f:\mathbb{R}\rightarrow\mathbb{R}$ by letting $f(x)=1-6x^2+4x^3$ for $0\leq x\leq 1$, and requiring that $f(x+2)=f(x)$ and $f(-x)=f(x) \ \ \forall x\in\mathbb{R}$. Find the Fourier series $Sf(x)$ [Hint: consider $f'(0), f '(1),f '''(x)$].
This question is part (b) of a question. For part (a), I was required to graph $f(x)$ for $-2\leq x\leq 3$. I believe I have done this correctly, but am wondering if this can help solve the above problem.
I'm unsure of how to properly utilise the 'hint' to find $Sf(x)$. I have calculated $$f '(0)=0$$ $$f '(1)=0$$ $$f ''' (x)=24$$ How does this help to find the Fourier Series of f?