1. The conflict in "The Outsiders" is between the Greasers and the Socs. The climax of The Outsiders is when Johnny kills the "Soc". Johnny and Ponyboy run away. Later, the Greasers and Soc's have another rumble, and Johnny dies. I would say that this is the climax.
2. ""We won," Dally panted. "We beat the Socs. We stomped them--- chased them outa our territory.""
"Johnny didn't even try to grin at him. "Useless... fighting's no good...." He was
awful white."
3. "But I had both arms wrapped around the other Soc's neck and refused to let go. Dally was slugging him, and I hung on desperately, although that other Soc was kicking me and you'd better believe it hurt. Finally he kicked me in the head so hard it stunned me, and I lay limp, trying to clear my mind and keep from blacking out."
"You read about people looking peacefully asleep when they're dead, but they
don't. Johnny just looked dead. Like a candle with the flame gone. I tried to say
something, but I couldn't make a sound."
4. The dialogue and textual evidence that I used to support my identification of the climax is taken from Chapter 9 in "The Outsiders". When reading the excerpts and reading the characters' dialogues, you can see exactly how tense and dramatic the section is. It is where most of the big dramatic events happen in the novel. With all of the drama, death, and fighting, there is no other part of the story that would be a better climax.
5. After the Greasers win the rumble, Dally dies. Ponyboy spends the rest of the story recovering after all of the physical and emotional trauma he went through. It was probably super scarring, dealing with his friends deaths as well as all of the physical pain that Ponyboy ended up enduring. I suppose that the climax relates to the falling action because the climax causes the falling action.
I really hope that this helps :)