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!"