Second Derivative Test

In calculus of single variable we applied the Bolzano-Weierstrass theorem to prove the existence of
maxima and minima of a continuous function on a closed bounded interval. Moreover, we developed
¯rst and second derivative tests for local maxima and minima. In this lecture we will see a similar
theory for functions of several variables.
De¯nition : A non-empty subset D of Rn is said to be closed if a sequence in D converges then
its limit point lies in D.
For example, the sets B1 = fX0 2 R2 : k X k· 1g and H = f(x; y) 2 R2 : x ¸ 0; y ¸ 0g are
closed subsets of R2. However, the sets S1 = fX 2 R2 : k X k < 1g and H+ = f(x; y) 2 R2 : x >
0; y > 0g are not closed.
De¯nition : Let D µ Rn and X0 2 D. We say that X0 is an interior point of D if there exists
r > 0 such that the neighborhood Nr(X0) = fX 2 Rn : k X0 ? X k < rg is contained in D.
For example, all the points of S1 are interior points of B1. Similarly, all the points of H+ are
interior points of H.
The notions of maxima, minima, local maxima and local minima are similar to the ones de¯ned
for the functions of one variable. The proof of the following theorem is similar to the proof of the
existence of maximum and minimum of a continuous function on a closed bounded interval.
Theorem 30.1(Existence of Maxima and Minima): Let D be a closed and bounded subset of
R2 and f : D ! R be continuous. Then f has a maximum and a minimum in D.