一般用the key to doing sth，意思是做.的关键
一般不用the key to do，但确实有这种结构.意思是开锁的钥匙，比如：I took out the key to unlock the door.(=I took out the key and unlocked the door.)我拿出钥匙打开了锁.不定式做目的状语
是的，句子中的 the key to keeping 不能换成 the key to keep，因为这样用的 to 是介词，不是不定式符号，它后面接动词时要用动名词，不能用动词原形。又如：
Language is the key to understanding those around you. 语言是与周围的人沟通的关键。
The driver of the car probably holds the key to solving the crime. 这位汽车司机很可能掌握侦破这一罪案的关键证据。
The humble potato may be the key to feeding the world’s population. 小小的马铃薯却可能担起养活全世界人口的重任。
Good product management is the key to gaining a foothold in the market. 良好的产品管理是占有市场一席之地的关键。
