To put forth one's utmost effort in English can be translated as "to give it one's all" or "to do one's best". This phrase likely comes from the idea of exerting all of one's energy, strength, or effort into a task or activity.

For example:

I will give it my all to make sure this project is a success.

我会竭尽全力让这个项目取得成功。

She always does her best in everything she does.

她在做任何事情时总是尽力而为。