べきだ and べきではない: appropriate actions and actions to avoid
Use べきだ to say what ought to be done according to a standard or judgment. Explore its negative form, retrospective regret, and the strength of the advice a listener receives.
What is the appropriate action?
べきだ follows a dictionary-form verb to present an action as something that ought to be done. 大切なことは自分で確認するべきです。 says that you should check important matters yourself. With する, both するべき and すべき are possible.
This is not simply a form for a schedule or prediction. 電車は九時に来るはずです。 is an expectation based on information, whereas 時間を守るべきです。 presents punctuality as the appropriate standard.
Negative judgments and looking back
べきではない strongly judges an action to be something one should avoid. 人の失敗を笑うべきではありません。 means that you should not laugh at other people's failures, not merely that laughing is optional or unnecessary.
出発前に確認すべきでした。 means that the speaker should have checked before leaving. In this context, it regrets an action that was not taken; it is not a report of an actual completed check like 確認しました.
The weight of advice for the listener
友達に謝るべきです。 directly states the judgment that apologizing to one's friend is the right thing to do. Because the speaker sets a standard, this carries more weight than a casual suggestion or a question about someone's preference.
For a simple recommendation, 少し休んだほうがいいですよ。 can suggest taking a short rest. Adding polite です does not remove the force of the judgment in べき, so consider the relationship and situation.
Quick check: Does ここで諦めるべきではありません。 mean that giving up is optional?
Show the answer and why
No. It judges giving up at this point to be something one should not do. The action is discouraged, not merely described as unnecessary.