I have been struggling trying to understand the difference between a quasiconcave and a convex utility function. As far as I understand a function can be both at a certain point, but is not clear to me why sometimes is said that the function is quasiconcave rather than just convex.

Can someone please tell me what are the implication of

Strict quasiconcavity implies single-peakedness, i.e. any strictly quasiconcave function has a unique supremum (or maximum if the domain is compact). Hence, any strictly increase convex function is also strictly quasiconcave.

Here are a couple figures to illustrate the difference. The function on the right is not quasiconcave because it has two local maxima.

enter image description here

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