TITLE: How to determine whether a function is concave, convex, quasi-concave and quasi-convex
QUESTION [8 upvotes]: I would like to know how to determine these following functions are concave or convex, and quasi-concave or quasi-convex:
$f(x) = 3 \text{e}^{x} + 5x^{4} - \text{ln}(x)$ and $f(x,y)=xy$.
Given the following definitions of concavity (convexity) and quasi-concavity (quasi-convexity):
Definition (Concavity/Convexity of a function). Let $f: \mathbb{R}^{n}\rightarrow \mathbb{R}$. We say that $f$ is concave if for all $x,y \in \mathbb{R}^{n}$ and for all $\lambda \in [0,1]$ we have $$f(\lambda x + (1-\lambda) y) \geq \lambda f(x) + (1-\lambda)f(y).$$ And a function is convex if $-f$ is concave, or $$f(\lambda x + (1-\lambda) y) \leq \lambda f(x) + (1-\lambda)f(y).$$
Definition (Quasi-concave/Quasi-convex). Let $f: \mathbb{R}^{n}\rightarrow \mathbb{R}$. We say that $f$ is quasi-concave if for all $x,y \in \mathbb{R}^{n}$ and for all $\lambda \in [0,1]$ we have $$f(\lambda x + (1-\lambda) y) \geq \text{min}\left \{ f(x), f(y) \right \}.$$ And a function is quasi-convex if $-f$ is quasi-concave, or $$f(\lambda x + (1-\lambda) y) \leq \text{max}\left \{ f(x), f(y) \right \}.$$ 
Concavity (Convexity) implies quasi-concavity (quasi-convexity) but not the other way around.
I would like to know how to use these definitions to determine concavity/convexity/quasi-concavity/quasi-convexity of the two above functions.
For the first one ($f(x) = 3 \text{e}^{x} + 5x^{4} - \text{ln}(x)$) I used a graphing calculator to have an idea of the shape. But that didn't help me. I wanted to take divide the function into parts as well.
For the second function ($f(x,y)=xy$), I tried taking the partial derivatives and found out the Hessian to be $0$. What does it mean?

REPLY [2 votes]: Given the generality of a function being merely quasi convex- a set of necessary conditions can be given in terms, when the function is differentiable see 
Otherwise for quasi convexity quasi concavity one just use the definitions.
Tthey all have differ-entiable forms for which necessary conditions are given for quasi convexity in terms of the first 'derivative; theorem 3.52 pager 67 in
http://link.springer.com/book/10.1007%2F978-3-540-70876-6
These will allow you to rule out whether a function is one of the two 'quasi's; once you know that the function is convex; one can apply the condition for quasi-linearity. If its convex but not quasi-linear, then it cannot be quasi-concave.
Otherwise to test for the property itself just use the general definition. Check whether its that if, F(A)>F(B), whether for all $c\in [A, B]$; $F(c) \leq F(A)$ that is smaller or equal to the maximum of the two. 
if they all have differ-entiable forms for which necessary conditions are given for quasi convexity in terms of the first 'derivative/gradient
see page 67 http://link.springer.com/book/10.1007%2F978-3-540-70876-6? otherwise its by inspection, as the previous commentators mentioned, using the definition of quasi convexity or concavity