Strictly concave

It can be shown from a Taylor series expansion that if/(x) has continuous second partial derivatives, /(x) is concave if and only if its Hessian matrix is negative-semidefinite. For/(x) to be strictly concave, H must be negative-definite. For /(x) to be convex H(x) must be positive-semidefinite and for/(x) to be strictly convex, H(x) must be positive-definite. 

Remark 5 The above two theorems can be directly extended to strictly convex and strictly concave functions by replacing the inequalities > and < with strict inequalities > and <. 

With usual assumptions placed on p and C(Qp) increasing and strictly concave, C increasing and strictly convex in both arguments), these marginal conditions are bo th necessary and sufficient to find the optimum. 

In Sects. 3.8 and 4.7, certain properties of concave functions [13-15] are needed. At the same time, properties of convex functions are obtained because the convex functions differ from the concave by the sign only and therefore their properties are mostly obtainable by reversion of all following inequalities. For simplicity, we do not restrict the domain of independent variables (although possible restriction on some concave subset, i.e., having property (A.71) below, is nearly obvious) and we confine ourselves to strict concave functions which suffice for our purposes. 

In order that there be values of C for which (5.41) has solutions that describe traveling necks and also periodic solutions, it suffices that g have a single turning point X (V) with dg(X)/dX > 0 for X < X (V) and with dg(X)/dX < 0 for X > X (V) clearly, there are values of V which give g this property whenever T is monotone increasing and strictly concave. Similarly, when T is a strictly convex increasing function, one can find values of V... 

Obviously, if /(x, a is (strictly) convex function, then /(x, a is (strictly) concave function. 

However, in many real scenarios, it has been observed that SLC s are risk averse with a strictly concave utility function. Equation 5. This is due to NDA being less sensitive to risk and SLC s... 

Conversely, a function f x) is strictly concave if —f x) is strictly convex. 

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