# Convex and Concave Functions

Definition: A function $f : I \to \mathbb{R}$ is said to be Convex if for every $x, y \in I$ and for every $t \in [0, 1]$ we have that $f(tx + (1 - t)y) \leq tf(x) + (1 - t)f(y)$. A function $f : I \to \mathbb{R}$ is said to be Concave if for every $x, y \in I$ and for every $t \in [0, 1]$ we have that $f(tx + (1 - t)y) \geq tf(x) + (1 - t)f(y)$. |

We now give equivalent definitions for convex and concave functions.

Theorem 1: Let $f : I \to \mathbb{R}$.a) $f$ is convex on $I$ if and only if for all $a, b, c \in I$ with $a < b < c$ we have that $\displaystyle{\frac{f(b) - f(a)}{b - a} \leq \frac{f(c) - f(a)}{c - a}}$.b) $f$ is concave on $I$ if and only if for all $a, b, c \in I$ with $a < b < c$ we have that $\displaystyle{\frac{f(b) - f(a)}{b - a} \geq \frac{f(c) - f(a)}{c - a}}$. |

*We only prove (a) above. The proof of (b) is analogous.*

**Proof of a):**Let $a, b, c \in I$ be such that $a < b < c$.

- $\Rightarrow$ Suppose that $f$ is convex on $I$. Then for all $t \in [0, 1]$ we have that:

- Set $a = x$, $b = tx + (1-t)y$, and $c = y$. Combining the first and third equations with the second equation gives us:

- Solving for $t$ gives us:

- Therefore:

- Similarly, we compute $1 - t$ to be:

- From the convexity of $f$ we have $f(tx + (1-t)y) \leq tf(x) + (1-t)f(y)$, or equivalently:

- And hence:

- Therefore:

- $\Leftarrow$ Obtained by working backwards from above. $\blacksquare$

We state yet another important definition for convex and concave functions.

Theorem 2: Let $f : I \to \mathbb{R}$.a) $f$ is convex on $I$ if and only if for all $a, b, c \in I$ with $a < b < c$ we have that $\displaystyle{\frac{f(b) - f(a)}{b - a} \leq \frac{f(c) - f(b)}{c - b}}$.b) $f$ is concave on $I$ if and only if for all $a, b, c \in I$ with $a < b < c$ we have that $\displaystyle{\frac{f(b) - f(a)}{b - a} \geq \frac{f(c) - f(b)}{c - b}}$. |

Theorem 2 gives us a nice characterization of convex functions. It tells us that a function $f : I \to \mathbb{R}$ is convex if and only if whenever we take three points $a, b, c \in I$ with $a < b < c$ we have that the slope of the line connecting $(a, f(a))$ and $(b, f(b))$ is less than or equal to the sope of the line connecting $(b, f(b))$ and $(c, f(c))$. In other words, the slope of the line segments connecting consecutive pairs of points on the graph of $f$ is increasing.

We can combine theorems 1 and 2 to get a nice chain of inequalities. That is, $f : I \to \mathbb{R}$ is convex if and only if for all $a, b, c \in I$ with $a < b < c$ we have that:

(9)