Question

What is the climax of the story, the point at which a decision is made that sets the outcome of the story?The youth and the princess are discovered to be in love.
The narrator describes the history of the arena.

The princess gestures to the youth to choose the door on the right.

The youth glances at the princess, asking her which door to choose.

Answer 1

The last option would be the climax, because it is the turning point of the story, it is when the youth has to make a decision. 
(Note: If this is from an actual story, then I haven't read it, this is just me using previous knowledge to give you an answer and advice) 
Hope this helps! :) 

Answer 2

"a decision is made that sets the outcome of the story"

The last option; "The youth glances at the princess, asking her which door to choose."

Would be the decision that is made that set the outcome of the remaining story. Based of the questions definition of "climax" . I'd assume that when reading that part of the story it created some type of built up suspense. The third option is what the youth decided to pick, but the last one make more sense to choose. 

Hope this helps.