Unconstrained local necessary conditions

In the unconstrained casef is deÖned on an open set. Use the

"variations"g(t)$f(x 0 +th)and chain rule.

Theorem

If x 0 is a local minimum then

(^10) =fx^0 (x 0 )=(∂f/∂x 1 (x 0 ),.. .,∂f/∂xn(x 0 )).
(^20) hTfxx^00 (x 0 )h=∑
i

∑

j

∂^2 f/∂xi∂xj(x 0 )hihj

Observe that one needs just twice di§erentiability atx 0.