x f(x) f'(x) g(x) g'(x)

1 6 4 2 5

2 9 2 3 1

3 10 -4 4 2

4 -1 3 6 7

$\displaystyle h(x) = f(g(x))-6$

The functions f & g are differentiable for all reals and g is increasing.

A) Explain why there MUST be a value r for $\displaystyle 1 < r < 3$ such that $\displaystyle h(r) = -5

$

b) Explain why c for $\displaystyle 1 < c < 3$ such that $\displaystyle h'(c) = -5$

I don't get how the value theorems tie into this problem.