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Choose the option in which the usage of the word is incorrect or inappropriate: face

A. The politician refused to lose face by admitting his error, despite mounting pressure.
B. It is crucial for leaders to face up to the difficult truths, even when unpopular.
C. The investors took the CEO's optimistic projections at the face value, much to their later regret.
D. In the face of severe budgetary constraints, the department had to re-evaluate its priorities.

The incorrect or inappropriate usage of the word "face" is in the third option: "The investors took the CEO's optimistic projections at the face value, much to their later regret."
Why it's wrong:
The standard idiomatic expression is "at face value," which means to accept something at its apparent worth or meaning without further investigation. The inclusion of the definite article "the" before "face value" is grammatically incorrect in this idiom.
Correct usage: "The investors took the CEO's optimistic projections at face value, much to their later regret."