I understand the definitions, and can do work with them, but when I try to picture them I get confused (picture simple concave/convex functions that is, not some very complex ones obviously).
Basically, I don't see why the definition for, say, convex functions of $f(\lambda x + (1-\lambda ) y) \leq \lambda f(x) + (1-\lambda )f(y)$ leads to pictures such as the one at the top of this mathworld wolfram page. If somebody could help me understand it better I would appreciate it.
Also, when looking at an equation for a function, are there any nice tricks to tell, without graphing, whether it is (quasi) convex/concave, or do I simply have to prove it or search for a counterexample? (This second question is similar to the one here, so I'm assuming the answer is no.)
Note: I mean... I kind of get it. Because of the $\leq$ we need to have a straight line between any two points lie above the function applied to the combinations that generate the points along that line, which would give us the kind of curvature in the picture, but then do all convex functions have to look like the one in the wolfram page?