In order find out the sentence with a linking verb, we must understand what linking verbs are. Linking verbs are used to link the subject with it's complement/state or adjective. For example: Roshni is pale. 'Roshni' is the subject and 'pale' defines her state, these two are connected by a linking verb 'is'.
Option A - Narayan looked at the window. Looked is a linking verb only when it links the subject Narayan with his state of being or an adjective that describes him. For example: Narayan looked pale. In this sentence, looked is an ordinary verb as it doesn't link the given subject with any complement. Hence, option A is incorrect.
Option B - Narayan tasted the food. Tasted is a linking verb only when it links the subject with its complement. For example: The juice tasted sweet. In this sentence, tasted is an ordinary verb as it doesn't link the given subject with any complement. Hence, option B is incorrect.
Option C - Narayan appeared in the room. Appeared is a linking verb only when it links the subject with its complement. For example: Narayan appeared calm, here, Narayans state of being is calm which is linked by appeared. In this sentence, appeared is an ordinary verb as it doesn't link the given subject with any complement. Hence, option C is incorrect.
Option D - Narayan remained calm. Remained is a linking verb, as it links the subject Narayan with its complement calm. Hence, option D is the correct answer.