If by maximum you mean the pressure recommendation by the car manufacturer, then yeah it will help with fuel saving. If by maximum you mean the pressure provided by the tyre manufacturer, please don't do that lol. Jumping from 26psi to 32psi isn't much, but jumping from 26psi to 49psi is way too much especially for car with only 1 passenger, it's gonna wear the middle of the tyre without the weight pressing down to give it an even wear. The money you save here will be used to replace the tyre earlier than it should. Will also cause terrible braking.

It gives you a fairly small gain in fuel mileage, but comes with several downsides.

For one, it's harder on your suspension system and gives you a rougher ride. This will wear out your struts faster, which is kind of expensive.

Next, it lowers the footprint of the tire with the ground. This means that you have decreased road grip and it will take longer to stop in an emergency.

Lastly, it means your tires won't last as long. The tread will wear uneven and it may cause tires that should last you 50,000 miles to only last 35,000, for instance. A rather large waste of money due to how expensive tires are, now.

Because on OPs car the recommended tire pressure is lower than the minimum recommended by the tire manufacturer. Increasing it even further may affect ride quality. In most scenarios your approach is correct.