The functions $f$ and $g$ are differentiable for all real numbers, and $g$ is strictly increasing. The table below gives values of the functions and their first derivatives at selected values of $x$.

$x$	$f(x)$	$f^{\prime}(x)$	$g(x)$	$g^{\prime}(x)$
1	6	4	2	5
2	9	2	3	1
3	10	-4	4	2
4	-1	3	6	7

The function $h$ is given by $h(x) = f(g(x)) - 6$.
(a) Explain why there must be a value $r$ for $1 < r < 3$ such that $h(r) = -5$.
(b) Explain why there must be a value $c$ for $1 < c < 3$ such that $h^{\prime}(c) = -5$.
(c) Let $w$ be the function given by $w(x) = \int_{1}^{g(x)} f(t)\, dt$. Find the value of $w^{\prime}(3)$.
(d) If $g^{-1}$ is the inverse function of $g$, write an equation for the line tangent to the graph of $y = g^{-1}(x)$ at $x = 2$.