SMILED at. Reading the sentence we find that:The incorrect preposition on is used along with the incorrect verb smile. On is a preposition that is used to denote the position of somethingTherefore it cannot be used in this sentence The correct preposition here is at. At is a preposition that expresses the object of a look, thought, action or plan. Here the object for the CHILD's smile is the mother and the preposition at denotes that.Example: I looked at my watch. The correct verb here should be smiled.As per the CONTEXT of the question, there are two possibilities. The statement can be in the past tense or in the present tense. Since the word child is SINGULAR it will be: The child smiles at her mother. But this option is not given in the options. Therefore we can go with the second option of the past tense. I.e using the past tense verb smiled. The correct sentence will be: The child smiled at her mother.