Despite the challenges, she remained ____________ about her future prospects.

The correct choice is "optimistic," which is an adjective used to describe a positive and hopeful attitude towards future events or outcomes. In the context of the sentence, the word "remained" is followed by a descriptive term that characterizes how she feels about her future prospects. "Optimistic" fits perfectly as it conveys that she holds a positive outlook despite facing difficulties.

Using "optimism," which is a noun, would not fit grammatically in this context where an adjective is required to describe her state of mind. The word "pessimistic," while it does describe a negative outlook, conveys the opposite sentiment of what is needed here. "Optimistically" is an adverb and would incorrectly modify a verb rather than provide a descriptive quality to the subject's outlook. Thus, "optimistic" is the only option that correctly completes the sentence, ensuring it conveys the intended meaning clearly and accurately.