Introduction 1: The function $ z(x,y) $ is a function of two variables. Hence, we cannot speak of a the derivative of $ z(x,y) $, but we have to specify whether we mean the partial derivative with respect to $ x $ or to $ y $, hence $ z'_x(x,y) $ or $ z'_y(x,y) $.

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 $ z(x,y) $, hence $ z''_{xx}(x,y) $, $ z''_{yy}(x,y) $ and $ z''_{xy}(x,y)=z''_{yx}(x,y) $.

Introduction 3: Whether a function of two variables is convex or concave depends on the sign of the criterion function
$$C(x,y)=z''_{xx}(x,y)z''_{yy}(x,y)-(z''_{xy}(x,y))^2.$$


Theorem:
  • If $ C(x,y)\geq 0 $, $ z_{xx}''(x,y)\geq 0 $ and $ z_{yy}''(x,y)\geq 0 $ on a part of the domain, then the function $ z(x,y) $ is convex on that part of the domain.
  • If $ C(x,y)\geq 0 $, $ z_{xx}''(x,y)\leq 0 $ and $ z_{yy}''(x,y)\leq 0 $ on a part of the domain, then the function $ z(x,y) $ is concave on that part of the domain.

A function such that $ C(x,y)< 0 $ on part of the domain, is neither convex nor concave on that part of the domain.