Let $g$ be the piecewise-linear function defined on $[-2\pi, 4\pi]$ whose graph is given above, and let $f(x) = g(x) - \cos\left(\dfrac{x}{2}\right)$.
(a) Find $\int_{-2\pi}^{4\pi} f(x)\, dx$. Show the computations that lead to your answer.
(b) Find all $x$-values in the open interval $(-2\pi, 4\pi)$ for which $f$ has a critical point.
(c) Let $h(x) = \int_{0}^{3x} g(t)\, dt$. Find $h^{\prime}\!\left(-\dfrac{\pi}{3}\right)$.