I have come across the following set of problems in my summer reading:
Prove that:
If f is strictly concave on the convex set K, then it is also strictly quasi -concave.
If t is an increasing function, and if f is a strictly quasi-concave function on a convex domain, then t o f is strictly quasi-concave on the same convex domain. (In other words, the property of being strictly quasi-concave is preserved under increasing transformations.)
Show that concavity is not necessarily preserved under increasing transformations.
Point 1 follows straight from $ \lambda f(x) + (1 - \lambda ) f(y) > min[f(x), f(y)] $, point 2 from t[min[f(x), f(y)]] = min[t(f(x)), t(f(y))] and point 3 can be shown by an example, for instance f(x) = ln(x) transformed by t(z) = exp(z)^2.
However, what I cannot get my head around is how point 2 and 3, especially in the light of 1, can be consistent with eachother! I have tried to think about it graphically but have not come across any solution. Does somebody have an intuitive explanation? Thank you!