A) to prevent the Soviets from gaining a military advantage.
B) to fufill requirements for joining NATO.
C) to provide defense for the United States' allies around the world.
D) to provide hope to humankind.
Soon after the USSR developed and successfully tested their first atomic device in 1949, Truman had a decision to make. The US made an initial tentative offer for both of the superpowers to turn over their few nuclear weapons to the United Nations, before a dangerous and expensive arms race ensued. While naive, perhaps, Truman's only other alternative became deterrence through superior technology. The Soviets rejected the American offer and the decision to pursue a hydrogen thermonuclear device became a reality.
There are a number of proposed reasons for Truman's decision; but I should first point out that the Bomb had already been developed. Truman's decision was to use it.
The correct answer would appear to be A; to prevent the Soviets from gaining a military advantage. There was ample evidence that the Soviet's intentions were quite aggressive; and that President Roosevelt had made a serious mistake at Yalta by giving Stalin a free hand. Even the Japanese historians now say that the decision was made to bring the war to a very quick end and not allow a Soviet invasion of Japan, which could have tipped the scales in favor of the Soviets in the upcoming Cold War.
Of the choices that you have given us here, the best choice is A. The point of the development of the bomb was to ensure that the Russians did not gain an advantage. However, it is also possible to argue that C is a good answer.
The hydrogen bomb was developed by the US because the US was worried that the Soviet Union would catch up to it in the arms race. The US wanted to stay ahead of the USSR. But the reason for this was that the US wanted to be able to defend itself and its allies around the world. Therefore, you could argue that both of these answers are correct. I would pick A, though, because it seems more correct to say that the US was trying to keep the USSR from getting an advantage since the H-Bomb is not really a defensive weapon.