When one end of the metal rod is dipped in boiling water, the temperature at that end would be maximum. When the other end of the rod is dipped in melting ice, the temperature of that end would be minimum. Due to conduction, the temperature of the rod starts to increase from maximum temperature end to minimum temperature end. After some time, equilibrium is established with the surrounding.
When Steady state is reached, the temperature throughout the rod is nonuniform and constant. Temperature will be different at different points on the rod. Due to equilibrium the state of the rod does not change.
Thus, option (d) is the correct option.