Example 1. Find a point $M$ on the ellipse $\frac{x^{2}}{16}+\frac{y^{2}}{9}=1$ such that the sum of its distances to the fixed points $A(\sqrt{7}, 0), B(1,2)$, i.e., $|M A|+|M B|$, is maximized or minimized.
Solve $A(\sqrt{7}, 0)$ and $A^{\prime}(-\sqrt{7}, 0)$ as two foci (as shown in Figure 1). Connect $A^{\prime} B$ intersecting the ellipse at $M_{1}$ and $M^{\prime}, \quad$ then $\left|A M_{1}\right|+$ $\left|B M_{1}\right|=8-$ $\sqrt{12-2 \sqrt{7}}$ is the minimum value, while $\left|A M^{\prime}\right|+\left|B M^{\prime}\right|=8+\sqrt{12-2 \sqrt{7}}$ is the maximum value.