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?
Best Answer
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.