Concavity/Convexity, QuasiConcavity,QuasiConvexity.

Question

I am given the piecewise {x^2 for x>0 and x for x<=0} and I am asked to find out if it is Convex/Strictly Convex/QuasiConvex/Strictly QuasiConvex and the same 4 for Concavity.

This is more of a "Is my thought process right?" Because not having a teacher due to covid is making this very challenging. (I am mostly guessing on the Quasi's)

I believe it can not be convex because the line between X=-0.1 and X=0.1 is below the graph

I believe it can not be concave because the line between X=0 and X=0.1 is above the graph

I believe it can not be strictly convex because it is not convex

I believe it can not be strictly concave because it is not concave

I believe it is all 4 forms of QuasiComplex/Concave because both the ABS Min and Max are the endpoints, so no value C in between them will be Higher or Lower than both endpoints.

If I am wrong on any of these please let me know (Don't give me the answer right away please! I want to be able to figure it out!)

Answer 1

"Is my thought process right?"

Yes.

You can also easily prove that if a function is (strictly) monotonic then it is both (strictly) quasiconvex and quasiconcave.