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Nidhi taparia Best Answer
Firstly, I am going to answer this question through the options i, ii, iii, and iv, because there is a mistake in the a., b., c., and d. options. Option i is definitely true because tangency is a necessary condition for optimality. Option ii is also true because tangency is a necessary condition not a sufficient one in general. Option iii is also true. When preferences are convex, tangency is a sufficient condition.