I've been doing some work that involves the following issue:
Let f be a real-valued function defined over the positive quadrant of the plane. Suppose the directional derivative of f at (1,1) in the direction <-1,-1> is negative, and the directional derivative of f at (1,1) in the direction <-1,0> is positive. Can the function f be strictly globally concave?
My wild guess is that f cannot have a strictly concave shape because there seems to be a "kink" of some sort at (1,1), but I don't know how to confirm my intuition. Is there an easy way to prove or disprove this? Any help would be much appreciated!