What is the climax of The Hunger Games?

The climax of The Hunger Games is during the would-be double suicide of Katniss and Peeta. Previously, the Gamekeepers had announced that there could be two victors if they were from the same district. Since Katniss and Peeta are both from District 12, they believe they both can live if they are the last two tributes left standing. In reality, the Gamekeepers announced this rule change to get the two to pair off during the competition, leading to a better show for the viewers.
This rule change was a lie. It is later announced that there can only be one victor. Katniss has a stroke of genius at this point: if she and Peeta, two star-crossed lovers, were to commit suicide, the Gamekeepers would be in quite a predicament, as they would have no victor.
She says,

Without a victor, the whole thing would blow up in the Gamemakers' faces. They'd have failed the Capitol. Might possibly even be executed, slowly and painfully while the cameras broadcast it to every screen in the country.
If Peeta and I were both to die, or they thought we were . . .

This leads to the climax—when Katniss acts as if she is eating the berries, telling Peeta to trust her. The readers are left on tenterhooks, not knowing whether the tributes from District 12 will live or die. At the last minute, the Gamekeepers decide to go back to the rule that allowed two winners, and Katniss and Peeta become the victors, winning the Hunger Games.

The climax of a story is the point in which the conflict is at its peak. The tension is at its highest point, and it is a turning point in the story that moves the narrative from rising actions to falling actions. In The Hunger Games, the climax occurs when Katniss learns that only one tribute can be declared the winner. This means that either Peeta or Katniss must die. Katniss decides that it is better to leave the Gamemakers with no winner rather than Peeta and her fighting to the death. Her plan is to force the Gamemakers to declare them both winners.

We both know they have to have a victor.
Yes, they have to have a victor. Without a victor, the whole thing would blow up in the Gamemakers' faces.

Katniss announces that she and Peeta will eat the poisonous berries and commit suicide. They go so far as to the put the berries in their mouths. Just as readers fear the worst, an announcement blares across the arena that Katniss and Peeta are both winners.

The berries have just passed my lips when the trumpets begin to blare.
The frantic voice of Claudius Templesmith shouts above them. “Stop! Stop! Ladies and gentlemen, I am pleased to present the victors of the Seventy-fourth Hunger Games, Katniss Everdeen and Peeta Mellark! I give you—the tributes of District Twelve!”