relation between inversion temperature, critical temperature and boyle temperature

µJT = ∂T/∂p
At the inversion temperature the coefficient changes its sign.
For higher temperatures it is negative and the gas heats up due to expansion. For a temperature below the inversion temperature µJT is positive and the expanding gas cools down.
The importance of this temperature is in gas liquefaction. Processes like the Linde process for air liquefaction need a positive coefficient. Hence you can only operate at temperatures below the inversion temperature.