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「食{た}べる」 can be negated as 「食べない」. 「美{うつく}しい」 can be negated as 「美しくない」. etc.
etc.

Can you negate 「べき」as in:  
「明日は受験なので、勉強するべきです。」( _Since you are taking a test tomorrow, you should
study._ )  
「明日は受験なので、遊ぶべき___。」( _Since you are taking a test tomorrow, you **should not**
play around._)

Every part of speech in Japanese seems negatable, but I can't figure-out how
to negate 「べき」。「遊ばないべきです」 is not the answer I am looking for.
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words
How to negate "べき"? (ie. "should not")
There is a word that is not used so frequently in everyday speech but that
nonetheless performs this task: べからず. You use it in the same was as べき/べし. An
example on alc shows the following sentence:

> 自然を侮るべからず。  
>  Don't underestimate nature.

I am not a native of course, so I can't comment on precisely where you should
and shouldn't use it, but I'd say in most situations, unless you know it's OK
to use it, you should just opt for the simpler ～べきではない or ～しないこと or ～してはいけない
or something similar (aka see 非回答者's post). In general searching I see it also
used in the phrase 「べからず集」 to refer collectively to things you should not do
(like as a list).
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Most naturally, you can use one of the two below.

「[遊]{あそ}ぶべきでない。」

「遊ぶべきではない。」

The second one places more emphasis on the negation.
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べき is a fun part of modern Japanese grammar with odd rules probably best
explained using the historical explanation.

# History

べし used to be a 助動詞{じょどうし} (auxiliary).

**Connecting to things before it:** べし followed the 終止形{しゅうしけい} (conclusive
form) normally, and with ラ変{へん} verbs it followed the 連体形{れんたいけい} (adnominal
form). (NB: One could think of it as _u_ being part of the morpheme: _-ube-_.)

**Connecting to things after it:** べし inflected like an adjective:

> 未然形 ｏ べから  
>  連用形 べく べかり  
>  終止形 べし ｏ  
>  連体形 べき べかる  
>  已然形 べけれ ｏ  
>  命令形 ｏ ｏ

**Evolution:** べし did not evolve like a normal adjective would (i.e., become
べい and end be able to sentences), but instead did something weird, presumably
due to its status as a 助動詞 and not a 形容詞.

# Now

Currently, べき behaves like this:

**Connecting to things before it:** べき still behaves completely like a 助動詞 in
this sense -- you need a fully surfaced verb to connect it to:

> ｘ猫だべき  
>  ｘ猫なべき  
>  ｘ猫のべき  
>  ｘ猫べき  
>  ｏ猫であるべき
>
> ｘ熱いべき  
>  ｏ熱くあるべき

**Connecting to things after it:** This is the odd bit. べき behaves a lot like
同じ when connecting to things after it: For nominals, べき it connects directly.
When concluding sentences, it takes the copula (which can be ∅ in informal
speech, but I think most people think of the copula still being there).

> やるべきこと  
>  行くべきだ

(Side note: there is also べし still around for ending sentences which people
use when trying to sound archaic or pompous or whatever, but amusingly, some
people say べしだ which means some people treat it as a noun also.)

# Your question

As written in other answers, but now with reasons:

  * You cannot use ないべき, because べき needs to attach to a fully-surfaced verb, and there is none in ない (it is a 助動詞). なくある is not acceptable (because it is the 助動詞「ない」 not the 形容詞「無い」).
  * You can use the old way of negating べき, which is べからず, but of course you will sound archaic if just using it in normal speech, like you're saying a proverb or quoting something.
  * The new way of negating it is by negating the copula: べきでない／べきではない.

However, there is a lingering thing that needs to be discussed:

> You should not [eat the cake]. ⇒ You shouldn't eat the cake.  
>  You should [not eat the cake].

In English (at least in my opinion), there is a subtle ambiguity in focus
between "You should not eat the cake." One way to read it is that there is an
action, and you should not take it. The other way is that there is an action
of not doing something, which you should take. The former way can be
contracted but the latter cannot.

(If you can't tell the difference, consider the dialogue: "Should I eat the
cake, or should I not eat the cake?" "You should not eat the cake.")

This is of course very subtle. What's the situation in Japanese?

In Japanese, you use べきでない for both. Since the negation is all the way on the
outside, it sort of suggests the "You should not [eat the cake]." nuance, and
if there is a situation that really needs the "You should [not eat the cake]."
nuance, you don't have anything like ないべき to use. In the case where I really
care about such a nuance, I think I would use ない方がいい.
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