How to determine whether a function is concave, convex, quasi-concave and quasi-convex

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?

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The second is neither convex nor concave – that’s easy to determine simply by looking at it. Along the line $y=x$, it is convex as a 1D function; along the line $y=-x$ it is concave. It is neither quasi-convex nor quasi-concave: to show not quasi-concave, consider the points $x = (0, 1)$, $y = (-1, 0)$, so $f(x) = f(y) = 0$. Parametrise the function along that line segment by $\lambda$; then $f(\lambda) = \lambda (\lambda – 1) < 0 = \min \{ f(x), f(y) \}$. You can rotate to get non-quasi-convexity.

The first is convex but not concave, and it’s not quasi-concave. Examine the value of $f$ at the points $x=1/3, x=10, x=1$ to see that it’s not quasi-concave. It’s convex again by inspection or by showing that its second derivative is strictly positive.

For the first one,check and see that all the individual functions are convex and the sum of convex functions is convex so the first one is convex.

Also for the second one you can check along lines as illustrated

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

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 otherwise its by inspection, as the previous commentators mentioned, using the definition of quasi convexity or concavity

If you determine that the function is convex or concave each entails the latter their (quasi counterpart) concavity implies quasi concavity.
Likewise with convexity.

Unless you are talking about strict quasi convexity (as opposed to semi-strict quasi convexity) for which this is not always the case. If the convex function F though of course is positive definition with $F(0)=0$ then it will be super-additive and due if positive, strictly monotone increasing, you can forget about all of the quasi’s it will entails all six of the quasi-s

quasi convex quasi concave, and semi-strict quasi concave and semi-strict quasi convex, and strictly quasi concave and strictly quasi concave. In addition it will be strictly pseudo convex. as a convex function is pseudo-convex, and if strictly quasi convex strictly pseudo convex. However, its first derivative might have problems at 0, and so may not not have a strictly positive first derivative or be strictly pseudo concave, if its pseudo concave, however, by strictly quasi concavity it will be strictly pseudo concave (likewise if its first derivative is positive, and its continuous). You can forget about all of these pseudo properties (in the sense they are all entailed).

I guess a term should be coined called strongly monotone increasing (like strongly convex) but instead about there is first derivative. There is for analytic/holomorhic functions. But that is a different story univalent.

if non-negative instead, $F(0)=0$ it will be monotonic increasing and thus will be quasi concave and quasi convex,

IF the function is monotonic, on a real interval, then the function will be quasi convex and quasi concave, that is a sufficient condition, although not necessary for the function to be quasi linear( both quasi convex or quasi concave) so if the derivative

$$\forall (x)\in dom(F): F'(x) \geq 0 $$ or

$$ \forall x \in dom(F): F'(x) \leq 0$$.

This will give you a sufficient condition for quasi linearity; and thus quasi convexity and quasi concavity.

If the function is strictly monotonically, increasing I believe it entails all of the quasi-‘s (if am not mistaken). Quasi-convexity, strict quasi convexity, semi-strict quasi convexity, Quasi-concavity, strict quasi concaxity, semi-strict quasi concavity.

; They also aren’t linear functions, so you rule out these functions being both concave and convex. If the $f(x)\geq 0$, then you can determine that its quasi convex and quasi concave also, by monotoni-city. Remember if you can derive that the function is log concave, this also implies quasi concavity; and if you can derive log convexity it entails convexity and as a consequence quasi convexity