Stationary points, tangents and transformations of a quartic

Find the $x$-value of the stationary point of inflection of the graph of $y = g(x)$.

Find the coordinates of the local minimum of the graph of $y = g(x)$.

Find the average rate of change of $g(x)$ between $x = 0$ and $x = 2$.

Find the equations of the tangent lines to the graph of $y = g(x)$ that have a gradient of $-2$.

Describe a sequence of two transformations that maps the graph of $y = g(x)$ to the graph of $y = g(1 - x)$.

Let $a$ and $b$ be positive real numbers, such that the graph of $y = a \times g(x) + b$ has a local minimum on the $x$-axis.

Express $b$ in terms of $a$.

For each real number $k$, let $g_k$ be the function $g_k : \R \to \R, g_k(x) = (x^2 - k)(x + k)^2$.

Find all values of $k$ for which the graph of $y = g_k(x)$ has exactly one $x$-intercept.