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In my understanding, the following sentences are acceptable but I am not 100% confident.

Please give me your opinion.

1. He runs fastest.

(In this case, "fastest" is the superlative form of an adverb, and the definite article before "fastest" is not required though "He runs the fastest," is acceptable as well.)

2. He runs the fastest of all the boys in his class.

(In this case, the definite article before "fastest" is required, because it is followed by "of phrase.")

3. He is the fastest runner.

(In this case, the definite article before "fastest" is required, because "fastest" is an adjective.)

4. She is happiest when she is alone.

(In this case, the definite article before "happiest" is not required, because it is not a sentence comparing her with other people.)

5. This lake is deepest here.

(The similar to the reason mentioned in 4.)
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It is okay to leave out "the," but it is usually when we are not really comparing one thing or person to another, but to itself or that same person.

I am happiest at the piano. (comparing myself to me)
He is most polite in front of his grandmother.

Another use of "most" is to express the idea of "very."

That cake is most delicious!
Comments  
EnglishmavenIt is okay to leave out "the," but it is usually when we are not really comparing one thing or person to another, but to itself or that same person. I am happiest at the piano. (comparing myself to me) He is most polite in front of his grandmother.Another use of "most" is to express the idea of "very." That cake is most delicious!

Thank you.

Let me confirm one thing.
Is it possible to leave out "the" before "fastest" in the following sentence?

He runs the fastest of all the boys in his class.