For a particle executing S.H.M. the displacement x is given by x=Acos⁡ωt. identify the graph which represents the variation of potential energy (P.E.) as a function of timet and displacement x

Potential energy is minimum (in this case zero) at mean position (x = 0) and maximum at extreme position (x =±A) At time t = 0, x = A, hence potential should be maximum. Therefore graph I is correct. Further in graph III. Potential energy is minimum at x = 0, hence this is also correct.