Example 4 If $a>0, b>0$, and $H=$ $\min \left\{a, \frac{b}{a^{2}+b^{2}}\right\}$. Try to find $H_{\max }$.

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Solution: Since $H$ is the minimum of $a, \frac{b}{a^{2}+b^{2}}$, $H$ is not greater than the geometric mean of $a, \frac{b}{a^{2}+b^{2}}$, i.e.,
$$
H \leqslant \sqrt{\frac{a b}{a^{2}+b^{2}}} \leqslant \sqrt{\frac{a b}{2 a b}}=\sqrt{\frac{1}{2}}=\frac{\sqrt{2}}{2},
$$

Equality holds if and only if $a=b=\frac{\sqrt{2}}{2}$, hence $H_{\max }=\frac{\sqrt{2}}{2}$.