I know that gull wings are shaped how they are for maximum help with a decrease in induced drag and increase in lift, but how do they do it? This is my homework!!!
Gull wings are a solution to two problems.
Requirement for increased speed will require a big propeller. The problem would be the propeller tips would hit the ground. You can solve this by either having longer landing gear (aircraft sits higher) or pushing the engine further out infront.
The problem with longer landing-gear is it needs to be heavier to ensure sufficient strength. Pushing the engine out further increases the overall length of the fuselage, good for stability but not in a fighter which needs to make tight turns.
Looking at the corsair you can see how the forward part of the fuselage is quite long with the engine stuck on the end. I believe this was the most they could do without sacrificing maneuverability. To get additional clearance without a heavy landing gear the gull-wing brought the gear height to a more manageable length.
The second issue is the fact that the Corsair was intended for carrier landings, which meant the landing gear had to be strong. A long landing gear would need to be very heavy to have the required strength. The gull-wing allowed a shorter (lighter) landing-gear.