1.19: Cosine curve from a graph

The diagram shows part of the curve $y = a + 4\cos bx$, where $a$ and $b$ are positive constants. The curve meets the $y$-axis at the point $(0, 6)$ and the $x$-axis at the point $\left(\tfrac{\pi}{6}, 0\right)$. The curve meets the $x$-axis again at the point $P$ and has a minimum at the point $M$.