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 $x = 2$.