Figure 3

Graphs of \(\varphi(s)\) and \(\varphi_{\hat{s}_{0}}(s)\) in Example 5.2. \({\overline{s} = \sigma k}\) since \(s_{\rho}= \sigma k\). Since \(\lim_{s \to\overline{s}-}{\varphi(s)} = \infty\), we have \(\hat{s}_{\mathrm{max}} = s_{\rho}= \sigma k\) and \(s_{\mathrm{max}} = \overline{s} = \sigma k\) for every \({\hat{s}_{0} \in [ 0, \hat{s}_{\mathrm{max}} )}\). The interval between \(s_{\mathrm{min}}\) and \({s_{\mathrm{max}} = \sigma k}\), depicted as a thick line segment in the s-axis, represents the set \(\{ s \in [ 0, \overline{s} ) \mid\varphi(s) \geq \varphi_{\hat{s}_{0}}(s) \}\)