Mathematics for Economists

Unconstrained local su¢ cient conditions

Theorem

If f is twice di§erentiable at x 0 and

fx^0 (x 0 )= 0

and the second derivative is

(^1) positive deÖnite then x 0 is a local minimum of f,
(^2) negative deÖnite then x 0 is a local minimum of f.
Observe that one needs just twice di§erentiability atx 0.