auxiliary, conjunction
in order to; for the purpose of
Formal/literary conjunction used after the dictionary form of a verb to express purpose. Often seen in written or formal speech.
I study every day in order to enter university.
We invited an expert in order to solve the problem.
auxiliary, conjunction
Archaic/literary auxiliary equivalent to modern べきだ. Used mainly in set phrases or classical-style writing. The form べく is the adverbial form of べし.
It is hardly surprising, but he was late again.
Those were words that had to be said.
ために is the common, neutral way to express purpose. べく is more formal and literary.
Adverbial form of the classical auxiliary べし, which expressed obligation, intention, or probability. In modern Japanese, it survives in formal purpose expressions and a few set phrases.