= Solution
Differentiate the negative-root equation $F(\beta_-,\delta)=0$, keeping $r$ and $\sigma$ fixed. Its derivatives are $F_\delta=-\beta_-$ and
$$
F_\beta=\sigma^2\beta_-+r-\delta-\sigma^2/2=-\Delta<0.
$$
Thus $d\beta_-/d\delta=\beta_-/F_\beta=-\beta_-/\Delta>0$. Differentiating the trigger ratio then gives
$$
\boxed{\frac{dS^*}{d\delta}=-\frac{X}{(\beta_--1)^2}\frac{d\beta_-}{d\delta}
=\frac{X\beta_-}{\Delta(\beta_--1)^2}<0.}
$$
This is the <dividend-yield sensitivity of the perpetual put trigger>. A larger dividend yield lowers the risk-neutral drift of the ex-dividend <stock>; waiting for further price reductions becomes more attractive to the put holder, so exercise occurs at a lower price. As checks, at $\delta=0$ the negative root is $-2r/\sigma^2$, giving $S^*=2rX/(2r+\sigma^2)$. As $\delta\to\infty$, $\beta_-\to0-$ and $S^*\to0$.
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