How do I find out what the climax of the story is?
Climax is the turning point and the most intense point in a story. To decide where the climax of a story takes place, always look at when we know the outcome of the main conflict. The climax usually occurs just before this. It usually involves an important event, decision, or discovery that affects the final outcome of the story.
In "The Necklace", Mathilde spends her life trying to pay for a new necklace for Madame Forestier. Mathilde didn't tell her she had lost the necklace and replaced it with a necklace of real diamonds. Mathilde's discovery that the necklace was fake certainly affects Mathilde. If she had just told Madame Forestier when she first lost the necklace, her life would have been lived very differently.
The climax occurs at the end of the story, when Madame Forestier tells Mathilde that the lost necklace was actually a fake.
Here is the exchange between the two women:
You say that you bought a necklace of diamonds to replace mine?"
"Yes. You never noticed it, then! They were very similar."
And she smiled with a joy that was at once proud and ingenuous.
Madame Forestier, deeply moved, took her hands.
"Oh, my poor Mathilde! Why, my necklace was paste! It was worth at most only five hundred francs!"
If you think of the climax as being the turning point in the story, then discovering that the necklace is missing has to be the climax. At that point the Loisel's lives change completely as they suffer through ten years of poverty to repay the loans required to replace the missing necklace. At this point even Mathilde herself changes. She stops her whining and complaining. "She played her part, however, with sudden heroism." In other words, she rose to the occasion and did everything within her power and station in life to assist her husband in repaying the debt. Although the discovery, upon meeting Jeanne Forestier years later, that the necklace was a fake, seems to be the most heightened or emotional part of the story; it is the resolution/conclusion or denoument because it resolves the mystery of the necklace and reveals the irony of her long suffering.