Example 3 The line $l$ passes through a fixed point $P(a, b)$ in the first quadrant and intersects the positive halves of the two coordinate axes at points $A, B$ respectively. Find the minimum value of the line segment $|A B|$.

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As shown in Figure 1, let $P M \perp x$-axis at $M$, $P N \perp y$-axis at $N$, and $\angle O A B=\theta$. Then, we have $|A B|=|A P|+|P B|=\frac{b}{\sin \theta}+\frac{a}{\cos \theta}$. According to the theorem, we have
$$|A B|_{\min }=\left(a^{\frac{2}{3}}+b^{\frac{2}{3}}\right)^{\frac{3}{2}}$$