Example 5 Given that $x, y$ are two variables in the interval $(0,1)$, denote the minimum value of $2^{-x}, 2^{x-y}, 2^{y-1}$ as $F(x, y)$. Find the maximum value of $F(x, y)$.
$(1984$, Shanghai Middle School Mathematics Competition)
Solution: Since $F(x, y)$ is the minimum of $2^{-x}, 2^{x-y}, 2^{y-1}$, $F(x, y)$ is not greater than the geometric mean of these three, i.e.,
$$
F(x, y) \leqslant \sqrt[3]{2^{-x} \times 2^{x-y} \times 2^{y-1}}=2^{-\frac{1}{3}} .
$$

When $x=\frac{1}{3}, y=\frac{2}{3}$, the equality holds, therefore, the maximum value of $F(x, y)$ is $2^{-\frac{1}{3}}$.