Convexity plays an important role not only in mathematics, but also in physics, chemistry, biology and other sciences.

We’ve learned what convex sets are, and now we”ll introduce convex functions.

## Convex and concave function

**Definition:** Let $I$ be an interval in $\mathbb{R}$.

**a)** We say that a function $f: I \rightarrow \mathbb{R}$ is ** convex **on interval $I$ if for all $x, y \in I$ and for all $\alpha \in [0, 1]$

**$$f(\alpha x + (1 – \alpha) y) \leq \alpha f(x) + (1 – \alpha)f(y).$$**

In addition, if for all $x \neq y$ and for all $\alpha \in \left<0, 1\right>$ previous inequality is strict, we say that $f$ is a ** strictly convex function**.

**b)** If an inequality is reverse, we say that $f$ is a* concave function*.

If for all $x \neq y$ and for all $\alpha \in \left<0, 1\right>$ that inequality is strict, we say that $f$ is a ** strictly concave function**.

Also, notice that $f$ is concave if function $-f$ is convex.

Furthermore, domain of a convex function must be a convex set.

Moreover, the following characterization of a convex function is valid.

Continuous function $f$ is * convex* on $[a, b]$ if and only if for all $x, y \in [a, b]$

**$$f\left(\frac{x + y}{2}\right) \leq \frac{f(x) + f(y)}{2}.$$**

## Examples of convex functions

**Example 1: **Function $f: \mathbb{R} \rightarrow \mathbb{R}$, $f(x) = x^2$ is strictly convex on $\mathbb{R}$.

**Example 2: **Function $f: \mathbb{R^{+}}\rightarrow \mathbb{R}$, $f(x) = x ln x$ is a convex function.

**Example 3: **Function $f: \mathbb{R} \rightarrow \mathbb{R^{+}}$, $f(x) = \vert x \vert$ is a convex function.

## Geometric interpretation of convexity

Let $f: I \rightarrow \mathbb{R}$ be a convex function and $$T_{1} = (x_{1}, f(x_{1})), T_{2} = (x_{2}, f(x_{2})),$$ where $x_{1}, x_{2} \in I, x_{1} < x_{2}$. Furthermore, let $$T_{\alpha} = (\alpha x_{1} + (1 – \alpha)x_{2}, \alpha f(x_{1}) + (1 – \alpha)f(x_{2})), \alpha \in [0, 1]$$ be a point on the line segment $\overline{T_{1}T_{2}}$.

We have $$f(\alpha x_{1}+(1 – \alpha)x_{2}) \leq \alpha f(x_{1})+(1 – \alpha)f(x_{2}).$$

In other words, the line segment between any two points on the graph of the * convex* function lies

*or*

**above***. On the other hand, the line segment between any two points on the graph of the*

**on the graph***function lies*

**concave***the graph.*

**below**

## Characterizations of convex functions

Sometimes it is difficult to determine by definition whether some function is convex or not. For that reason, we often use characterizations of convex functions.

**Theorem: **Let $f: I \rightarrow \mathbb{R}$ be a derivable function on $I$. Function $f$ is ** convex** if and only if its derivative $f’$ is an increasing function.

**Corollary:** Let $f: I \rightarrow \mathbb{R}$ be two time derivable function on $I$. Function $f$ is * convex* on interval $I$ if and only if $f”(x) \geq 0$.

**Example: **Determine whether these functions are convex:

**a)** $f(x) = e^x$

**b)** $f(x) = x^p, x \geq 0$

**c)** $f(x) = ax + b, a, b \in \mathbb{R}$

**Solution:**

**a)** The second derivative of $f(x) = e^x$ is $f”(x) = e^x \geq 0$. In conclusion, it is a convex function.

**b) **The second derivative of $f(x) = x^p, x \geq 0$ is $f”(x) = p(p – 1)x^{p – 2}$.

$f”(x)$ is non – negative if $p(p – 1) \geq 0$. In other words, it is non – negative if $p \in \left<- \infty, 0\right] \cup \left[1, + \infty \right>$. Furthermore, it is negative if $p(p – 1) < 0$. In other words, it is negative if $p \in \left<0, 1\right>$.

In conclusion, $f$ is convex for $p \leq 0$ or $p \geq 1$ and concave for $0 \leq p \leq 1$.

**c)** Obviously, the second derivative of given function is $f”(x)\geq 0$. Therefore, $f(x) = ax + b$ is convex function, but also concave.