Arnol'd's second stability theorem is approached from an elementary point of view. First, a sufficient criterion for stability is found analytically as either -[mu][subscript 1] less than U[double prime]/U-U[subscript s] less than 0 or 0 less than U[double prime]/U-U[subscript s] in the flow, where U[subscript s] is the velocity at the inflection point, and [mu][subscript 1] is the eigenvalue of Poincare's problem. Second, this criterion is generalized to barotropic geophysical flows in the [beta] plane. And the connections between present criteria and Arnol'd's nonlinear criteria are also discussed. The proofs are completely elementary and so could be used to teach undergraduate students.