Suppose that a differentiable functional f on an open set S n is quasiconcave. At every regular point f(x0) 0, A restricted form of quasiconcavity

Suppose that a differentiable functional f on an open set S Š† „œn is quasiconcave. At every regular point ˆ‡f(x0) ‰  0,

A restricted form of quasiconcavity is useful in optimization (see section 5.4.3). A function is quasiconcave if it satisfies (37) at regular points of f. It is pseudoconcave if it satisfies (37) at all points of its domain. That is, a differentiable functional on an open convex set S ˆˆ „œn is pseudoconcave if

A function is pseudoconvex if -f is pseudoconcave. Pseudoconcave functions have two advantages over quasiconcave functions-every local optimum is a global optimum, and there is an easier second derivative test for pseudoconcave functions. Nearly all quasiconcave functions that we encounter are in fact pseudoconcave.

f(x) >f(xo) Vf(xo)' (x-xo) > 0 for every x, xo in S (37) f(x) >f(xo ) Vf(x), (x-xo) > 0 for every X, XO E S (38)