Let \(c(\nu,k,\alpha)\) be the k-th positive x-zero of \(\cos\alpha J_{\nu}(x)-\sin\alpha Y_{\nu}(x)\). Using a method due to the first author, Stud. Sci. Math. Hungar. 12, 81-88 (1977; Zbl 0435.33006), the second author and the reviewer, J. Math. Anal. Appl. 98, 470-477 (1984; Zbl 0549.33005) showed that, for \(k \geq 2\), \(c(\nu,k,\alpha)\) is a concave function of nu on \((0,\infty)\) and that \(c(\nu,l,\alpha)\) is concave on \((0,\infty)\) at least for \(0 \leq \alpha \leq \pi / 2\). In the present paper it is shown that this last result fails for \(\alpha\) close to and less than \(\pi\). In effect the authors show the existence of a number \(\epsilon\) (= 0.336697...) such that if \(\pi-\epsilon < \alpha < \pi\), then \(c(\nu,l,\alpha) \leq \nu + \epsilon - 1/2\) and \(c(\nu,l,\alpha)\) is a convex function of \(\nu\) on \([1/2,\infty)\).