Purpose: べく vs んがために, and the べき family
A small category, but the べく / べきだ / べからず trio share a root and are easy to misread. べく is adverbial purpose, べきだ is predicate obligation, べからず is classical prohibition.
The patterns
べく
in order to (written; attaches to dictionary form, する → すべく)
問題を解決すべく、対策を練った。
In order to solve the problem, we devised countermeasures.
んがため(に)
for the sake of (archaic-literary; ない-form minus い + んがため)
目的を達成せんがために、あらゆる手段を尽くした。
To achieve the goal, they exhausted every means.
べきだ
should; ought to (predicate, ends the sentence)
約束は守るべきだ。
You should keep your promises.
べからず
must not (classical prohibition, signs and formal text)
立ち入るべからず。
No entry.
Decision procedure
What role does the word play in the sentence?
For purpose: how archaic is the register?
Commonly confused
べく vs べきだ
Nearly identical in shape, entirely different in job: べく sits mid-sentence marking purpose, べきだ closes the sentence stating obligation. This is the pair actually worth memorising — べからず is rare enough to just recognise.
んがため attachment
It attaches to the ない-form minus い. する becomes せんがため, because する has three irregular imperfective stems (し / せ / さ) and this pattern takes せ.