Using the built-in EEPROM seems to be be a perfect fit to your problem. Read the documentation of the EEPROM library included with the IDE.
Use EEPROM.put to save Min and Max every time one of them is updated. Use EEPROM.get to restore the values in your setup() function.
How often do you need to reset the min and max values? Are they "eternal", i. e. "the highest temperature ever encountered since I installed this device" or do you need to reset the min and max, i. e. "the highest temperature since this morning when I restarted the device"? In the latter case you may want to provide a method to reset the EEPROM values. Maybe include a button input for "if button is held down while restarting then reset previous min and max values"