Properties of convex functions

Properties of Convex Functions A function $f(x)$ is said to be convex if the following property holds for all $x$ in the domain of $f$: 1. Monotonicity:...

Properties of Convex Functions

A function $f(x)$ is said to be convex if the following property holds for all $x$ in the domain of $f$ :

1. Monotonicity: If $a \leq b$ , then $f(a) \leq f(b)$ .

2. Concavity: If $f''(x) > 0$ for all $x$ in the domain of $f$ , then the graph of $f(x)$ is concave upwards.

If $f''(x) < 0$ for all $x$ in the domain of $f$ , then the graph of $f(x)$ is concave downwards.

3. Superposition: If $f(x_1) + f(x_2) \leq f(x_1) + f(x_2)$ for all $x_1$ and $x_2$ in the domain of $f$ , then the graph of $f(x)$ is convex.

4. Homogeneity: If $f(λx) = λf(x)$ for all $x$ in the domain of $f$ , then the graph of $f(x)$ is invariant under dilation.

5. Duality: If $g(x) = f(-x)$ , then $g'(x) = -f'(x)$