Example 2. On the parabola $y=2 x^{2}$, there is a moving point $A$ ( $m$, $\left.2 m^{2}\right), m \in\left[\frac{1}{2}, 1\right]$ and $B\left(n, 2 n^{2}\right)$. The tangent line at point $A$ is perpendicular to $A B$. Find the maximum and minimum values of $n$.
Solve: The slope of the tangent line to the parabola $y=2 x^{2}$ at $A\left(m, 2 m^{2}\right)$ is $4 m, A B$'s slope is $\frac{2 n^{2}-2 m^{2}}{n-m}$ $=2(n+m)$. From the condition $4 m \cdot 2(n+m)$ $=-1, n=-\left(m+\frac{1}{8 m}\right)$. Therefore, $n$ on $\left[-\frac{1}{2}, 1\right]$

has the maximum and minimum values of $-\frac{3}{4}$ and $_{8}^{5}$, respectively. (Note: Jiangsu Gaoyou County High School)