Sufficiency of Kuhn-Tucker

On page 212 of your latest updated slides, on the “sufficiency” of Kuhn-Tucker conditions, can the conclusion be true if we do not assume local non-satiation? If we do not assume monotonicity? strict monotonicity? It’s interesting if it is, because the would mean even if one of the two goods is a bad, or whatever pathological preferences we can think of, the Kuhn Tucker conditions are still necessary and sufficient.
Given a concave, differentiable utility function, the Kuhn-Tucker conditions are indeed sufficient to describe a solution to the UMP. Even if preferences are strange. Try a few examples with strange preferences and you’ll see!

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