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We study local and global optimality of geodesics in the left invariantsub-Riemannian problem on the Lie group $\mathrm{SH}(2)$. We obtain thecomplete description of the Maxwell points corresponding to the discretesymmetries of the vertical subsystem of the Hamiltonian system. An effectiveupper bound on the cut time is obtained in terms of the first Maxwell times. Westudy the local optimality of extremal trajectories and prove the lower andupper bounds on the first conjugate times. We also obtain the generic timeinterval for the $n$-th conjugate time which is important in the study ofsub-Riemannian wavefront. Based on our results of $n$-th conjugate time and$n$-th Maxwell time, we prove a generalization of Rolle's theorem that betweenany two consecutive Maxwell points, there is exactly one conjugate point alongany geodesic.

Maxwell Strata and Conjugate Points in the Sub-Riemannian Problem on the Lie Group SH(2)