For example, $2P$ is a moving point on the ellipse $\frac{x^{2}}{4}+y^{2}=1$, and $F_{1}, F_{2}$ are the left and right foci, respectively. Try to find the maximum and minimum values of $\left|P F_{1}\right| \cdot\left|P F_{2}\right|$.
(1996 "Hope Cup" Competition Question)
Solution 1: By property $5(\mathrm{I})$, it is known that the maximum value of $\left|P F_{1}\right| \cdot\left|P F_{2}\right|$ is 4, and the minimum value is 1.
Solution 2: By property 8, it is known that $\left|P F_{1}\right| \cdot\left|P F_{2}\right|=b^{2}+a^{2}-d^{2}=5-d^{2}$.
Furthermore, by the range of the ellipse, we know $b^{2} \leqslant d^{2} \leqslant a^{2}$, i.e., $1 \leqslant d^{2} \leqslant 4$, hence the maximum value of $\left|P F_{1}\right| \cdot\left|P F_{2}\right|$ is 4, and the minimum value is 1.