In the sentence:
Max wanted to cook dinner for his friends.
The underlined phrase which is ‘to cook dinner ‘ is an infinitive phrase. Infinitive phrases starts with infinitives that is, “to” added by a simple verb and can be used as nouns, adjectives or adverbs. In the case of including a main clause in the sentence, infinitive phrase separates it with comma. It is not participial because it (phrase) did not start with a participle and not a gerund because the first word of the phrase has no ‘-ing’.