STOCHASTIC

284 O. L. MANGASARIAN
We have from (2.4) through (2.7) that/(\) achieves its maximum at X.
Hence it follows by the differentiability of 0(x) and the chain rule that
(2.9) (x 2 - x l )'V, 6(S) = ^ - = 0.
Since
(2.10) x - x = x 2 - (1 - XV - Xx 2 = (1 - X)(x 2 - x 1 ),
it follows from (2.9) and (2.10) and the fact that X < 1, that
aX
(2.11) (x 2 - x)'V,9(x) = 0.
But by the pseudo-convexity of 0(x), (2.11) implies that
(2.12) 9(x 2 ) g S(x).
Hence from (2.1) and (2.12),
e(x l ) > e(x),
which contradicts (2.4). Hence 0(x) must be strictly quasi-convex on C.
That the converse is not necessarily true can be seen from the example
8(x) =x\ x 8(x),
which contradicts (2.13). Hence 8{x) £