Convex Functions, Concave Functions
Definition
Given an interval $I \subset \mathbb{R}$, two elements $x_{1} , x_{2}$ and functions $f : I \to \mathbb{R}$ and $0 \le t \le 1$,
- When $f( t x_{1} + (1-t) x_{2}) \le t f(x_{1}) + (1-t) f(x_{2})$, $f$ is defined as a convex function on $I$.
- When $f( t x_{1} + (1-t) x_{2}) \ge t f(x_{1}) + (1-t) f(x_{2})$, $f$ is defined as a concave function on $I$.
Explanation
Since there are too many confusing terms for convex or concave, whether it’s upwards convex or downwards concave, it’s strongly recommended to remember the graph shapes and correspond them with convex and concave, just as they are expressed in English. Although at first glance, the definition may seem unfamiliar just by looking at the formulas, thinking about the concept of internal division can make it a very intuitive definition to accept. It’s an intuitively simple concept, so there’s no need to memorize the definition if there’s no need for formulaic elaboration or explanation. Typically, starting from the quadratic functions in junior high school, constant exposure to the properties of second derivatives and their signs makes their properties familiar as well.
To be honest
To be honest, just assume that concave isn’t used much and it’s mostly convex.
Second Derivative
Second derivative of a convex function: Let’s say $f$ is twice differentiable on $I$. That $f$ is convex on $I$ and $f '' (x) \ge 0$ are necessary and sufficient conditions.
Note the added condition of being twice differentiable here. Usually, curves like $y = x^2$ or $y = \ln {x}$ are used as examples, making it easy to overlook, but as we have redefined the convex function, ‘continuous’ wasn’t even mentioned.