Question

show that heat and work are path functions

Answer 1

Heat (work) is a path function because how you move from one state to another determines the work done on or by the system. Thus, heat can't be a state function. For instance, if a gas expands isothermally, then heat has to be supplied to the system so that the gas maintains its temperature as it expands. But if you do this adiabatically, then the system does work. Same final state (pressure and volume) bt different work and heat.