VWO 1987

Find all continuous functions $f: \mathbb{R}\rightarrow\mathbb{R}$ such that $$f(x)^3 = -\frac x{12}\cdot\left(x^2+7x\cdot f(x)+16\cdot f(x)^2\right),\ \forall x \in \mathbb{R}.$$

Show that for $p>1$ we have $$\lim_{n\rightarrow+\infty}\frac{1^p+2^p+...+(n-1)^p+n^p+(n-1)^p+...+2^p+1^p}{n^2} = +\infty$$ Find the limit if $p=1$ .