〜べく
in order to (formal)
Verb dictionary form + べく (する → すべく or するべく)
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べく is a formal, written "in order to": 夢をかなえるべく、上京した, "I moved to Tokyo to pursue my dream".
It follows the dictionary form. With する, the classical すべく is common (解決すべく), though するべく is also used. The second half is a deliberate action the subject takes, so it's typical of news reports and formal statements: 原因を調べるべく、調査団が派遣された.
It's the adverbial form of べき (N3, "should"), so the feeling is "as one should, in order to". Don't mix them up: 行くべき人 is "a person who should go", but 行くべく is "in order to go".
The negative 〜べくもない (N1) means "can't possibly".
Examples
夢をかなえるべく、上京した。
I moved to Tokyo to pursue my dream.
問題を解決すべく、全力を尽くします。
We'll do everything we can to resolve the problem.
家族を養うべく、懸命に働いた。
He worked hard to support his family.
新記録を出すべく、毎日練習している。
She practises every day to set a new record.
事故の原因を調べるべく、調査団が派遣された。
An investigation team was sent to find the cause of the accident.