The correct option is C is
In order to fill in the third blank with the most appropriate linking verb, let's first understand what linking verbs are. Linking verbs are the words that link the subject to a noun or it's adjective. For example: I am a girl. Here, 'I' is linked to the word 'girl' by the word 'is', hence 'is' is the linking verb here.
Since the first and last sentence is in present tense, we must fill in the blank with the same tense.
Option B - 'Her health looking bad', is grammatically incorrect, as the participle form is always accompanied by helping or auxiliary verb. Though 'looking' is a linking verb, it doesn't link the given subject with its complement. Hence, option B is incorrect.
Option C - 'Her health have gone bad' is grammatically incorrect. 'Gone' is a past tense of 'go', it also means 'become' at times. 'Health' is a singular subject, hence the helping verb should be in the singular form. 'Health has' is correct according to the subject-verb agreement. Though 'gone' is a linking verb, it doesn't link the given subject with its complement. Hence, option C is incorrect.
Option D - 'Become' is used with plural subject and not singular; also, 'her health' is a singular subject. 'Her health become bad' is grammatically incorrect. Though 'become' is a linking verb, it doesn't link the given subject with its complement. Hence, option B is incorrect.
Option A - 'Her health is bad' is grammatically correct. As 'her health' is a singular subject, 'is' is used as a singular form of the verb. 'Is' is a linking verb that links the subject 'her health', with her complement 'bad'. Hence, option A is the correct answer. 'Her health is bad'.