Step 1: Understanding the Concept:
The context of this sentence highlights a person experiencing extreme exhaustion due to a "grueling schedule" and facing "sheer pressure."
The preposition "under" immediately following the blank is a critical structural clue.
We need a phrasal verb that naturally pairs with "under" to mean collapsing, surrendering, or breaking down due to mental or physical stress.
Step 2: Detailed Explanation:
Let’s look at how the options interact with the word "under":
1. Cave in (Option A): This means to collapse inward physically, or figuratively to yield, surrender, or break down under pressure.
The phrase "cave in under pressure" is a common idiomatic expression in English. It perfectly matches the context of being overwhelmed.
2. Buckle up (Option B): This means to fasten one’s seatbelt in a vehicle.
Note: If the option were "buckle under," it could mean to yield, but "buckle up under" is grammatically incorrect.
3. Give out (Option C): This means to break down, stop functioning, or be exhausted (e.g., "Her legs gave out").
While it matches the theme of exhaustion, it does not grammatically take the preposition "under" in this manner. You don't say "give out under the pressure" to mean surrender.
4. Tire out (Option D): To make someone completely exhausted.
This is a transitive verb phrase usually requiring an object (e.g., "The schedule tired her out"). Since "exhausted" is already in the sentence, this would be redundant.
Thus, "cave in" is the grammatically and contextually perfect selection.
Step 3: Final Answer:
The correct phrasal verb to fill the blank is "cave in."
Therefore, the correct option is (a).