Mathematics for Economists

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.