quasiconcave vs convex function
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.