Let $h(x) = \frac{x^4}{(1-x)^4}$ and $g(x) = f(h(x)) = h(x) + 1/h(x)$. Sketch the graph of $g(x)$ and show that $g(x)$ has a root between $0$ and $1$.
$$h'(x) = \frac{4x^3(1-x)^6 - 6(1-x)^5 x^4}{(1-x)^4}$$ $$g(x) = h(x) + 1/h(x)$$ From graph sketching theory, $x_0$ must be a solution of $h(x) + 1/h(x) = 0$ which lies between $0$ and $1$. It can be shown that $g(x)$ has a root between $0$ and $1$.
Let $h(x) = \frac{x^4}{(1-x)^4}$ and $g(x) = f(h(x)) = h(x) + 1/h(x)$. Sketch the graph of $g(x)$ and show that $g(x)$ has a root between $0$ and $1$.