# Second-order condition

Introduction 1: The function is a function of two variables. Hence, we cannot speak of a the derivative of , but we have to specify whether we mean the partial derivative with respect to or to , hence or .

Introduction 2: A function of one variable is convex if the derivative increases, hence if the second-order derivative is non-negative. A similar reasoning holds for functions of two variables. We have to consider, however, the second-order partial derivatives of , hence , and .

Introduction 3: Whether a function of two variables is convex or concave depends on the sign of the criterion function

Theorem:
• If , and on a part of the domain, then the function is convex on that part of the domain.
• If , and on a part of the domain, then the function is concave on that part of the domain.

A function such that on part of the domain, is neither convex nor concave on that part of the domain.