The highest point of a play or a novel or a short story is usually called the climax. That climax might be the most exciting part of the action. In that case, then you could say the climax is when Julie kills herself with Jean's razor.
However, the climax is the turning point of the play, after which all the action leads to a final resolution. In that case, then the climax of this play occurs when Julie follows Jean into his bedroom and the audience is left to their own imaginations. Julie regrets her decision because she never wanted to be controlled by a man, let alone a servant, from a lower social class than hers. On top of all of that, her mother raised her as a boy, so she was confused about gender and sexual identity. All of this leads to her tragic end.