Unconstrained local su¢ cient conditions
Asf is twice di§erentiable atx 0
f(x 0 +h) f(x 0 )=
fx^0 (x 0 ),h
+^1
2
hTfxx^00 (x 0 )h+o
khk^2
.
LetH=fxx^00 (x 0 )and letα>0 be the constant above. For anyε<α/ 2
one has that (^)
o
khk^2
(^)
εkhk^2
forhsmall enough. Hence asfx^0 (x 0 )= 0
f(x 0 +h) f(x 0 )
α
2 ε
khk^2 0.