This question points to one of the more curious elements of Williams' play, Sweet Bird of Youth. Though this is a well-crafted play with a series of definite conflicts, the climax wherein these conflicts reach their breaking points is not entirely obvious.
The play's primary conflict belongs to Chance and his character. He has come back to his home town where he is immediately threatened and told to leave. If Chance chooses to stay in order to regain his dignity and claim his identity as the beautiful and daring darling of the town, he may suffer physical harm. Yet, this is the only way he can maintain the only persona he can embrace. All other roads lead Chance to disreputability, or at least to anonymity.
A lesser conflict is connected to the Princess character. The actress exists in denial and in flight but is on the point of making a critical decision. She will either choose to stay with Chance and let go of her past success or she will attempt to return to Hollywood and to making films. This choice is essentially a choice between evasion and honesty.
In the second act of the play, Chance's story-line reaches its logical climax but not its emotional climax. His love interest, Heavenly, denies and rejects him. Chance then must find another way to salvage his dignity and self-esteem.
He does this by forcing the Princess to face her decision, pushing her to the climax of her conflict by calling her agent in California. This conversation will effectively decide the direction the Princess will take while also, Chance hopes, providing one last opportunity for Chance to regain his glorious potential, his status as a winner, and his opportunity to get Heavenly back.
The phone call does decide the fate of the Princess. Chance does not get what he wants from the phone call and this result reveals Chance's deepest conflict, which is now unrelated to the town and to love. Chance is forced to face the fact that his identity (persona) is a false one and, through this, forced to make the same decision that the Princess has made - to continue evading the truth or to be honest about it.
Aging and time are parts of life that cannot be avoided, but only Chance cannot accept that by the end of the play.
In this way we can argue that the climax of the play for both Chance and the Princess comes in the final act of the play, immediately after the phone call between the Princess and her agent.
The climax should be around Act 2 Scene 2, where Chance confronts Tom Junior and learns of Heavenly's hesrectomy. At the same time the heckler goes to confront Boss Finley.
Scene 3 should be the resolution of the play, where the Princess and Chance come to accept the end of their career/youth. Williams brings about resolution in the form of Chance facing his castration.