In the sentence الولدُ المجتهدُ ناجحٌ ("The hardworking boy is successful"), the word المجتهدُ functions as a نعت (adjective/qualifier) for الولدُ. A نعت is a word that describes a preceding noun (the منعوت) and must agree with it in four grammatical features: definiteness (التعريف والتنكير), case (الإعراب), gender (الجنس), and number (العدد). Here, المجتهدُ is definite (has ال), masculine, singular, and in the nominative case (مرفوع), matching الولدُ exactly.
The other options do not apply. حال (a circumstantial expression) describes the state of the subject or object during an action and is typically indefinite (e.g., جاء الولد مبتسمًا). فعل means "verb," but المجتهد is clearly a noun/adjective, not a verb. اسم إشارة (demonstrative pronoun) refers to words like هذا, هذه, ذلك, and تلك, which point to specific things, and المجتهد is not one of these.
Exam tip: The key test for نعت is agreement with the noun it follows in definiteness, case, gender, and number. If a definite adjective immediately follows a definite noun and matches it in all four features, it is almost certainly a نعت.