Answer: make the best use of.
Explanation:
The meaning of the word maximize in this case is: to use (something) in a way that will give the best result. The student mentions that he has limited amount of time for homework, so he wants to use the time available in a best possible way.
The synonyms of this verb that could be used in this sentence are: make the most of, make the best use of, etc. In other contexts, synonyms for 'maximize' are increase, make bigger, enlarge, etc.
