Tangents, area and stationary points of a rational function

State the coordinates of both stationary points of $f$.

Find the area bounded by the graph of $y = f(x)$, the line $x = -1$, the line $x = 1$ and the $x$-axis.

Find the equation of the tangent line to the graph of $y = f(x)$ at the origin.

Find the equations of the tangent lines to the graph of $y = f(x)$ that have a gradient of $-\frac{1}{8}$.

Let $a, b \in \R$, where $0 < a < b$.

Show, using algebra, that the average rate of change of $f$ over the interval $[a, b]$ is given by $\frac{1-ab}{(a^2+1)(b^2+1)}$.