A couple of thing here if I could. The MAX PSI on the side of a tire, any tire, is just that, the maximum pressure the tire is designed to hold. Tire pressure designated by the manufacture is a blend of the suspension. Your butt is isolated from the road by the seat, springs, shocks/struts, anti-sway bars, and the flexibility of the tires based on air pressure. A tire aired to the max will make your car ride like a red wagon. However, if you have standard tires and you want to do a little road racing then it is recommended to air them to the max so as to stiffen the side walls thus helping to avoid sidewall failure. Once the session is over the tire should be retuned to the recommendation on the door placard.
It should be noted that the auto manufacture’s tire pressure recommendation found on the door placard is based on the suite of OEM tires used on that specific vehicle. If you change to a different tire, especially a lower profile tire, then you should consult with the tire maker’s web site for pressure recommendations – usually (I said usually and not always) the folks at the tire store have no idea.
As noted herein, tires purged of air and fill with nitrogen will have more stable pressure. Lower profile tires will typically loose more pressure as the outside temperature drops and gain more pressure as outside temperature rises. Nitrogen will lessen, but not eliminate, these swings.