PROBLEMS 977
for 0 < x2 + "y2 K 2. Accordingly, f(x, y) > f(0, 0) for 0 < x2 + y2 < 82, and
f has a local minimum at (0, 0).
For A K 0 the proof that f has a local maximum is similar.
Case II. B2 - 4AC > 0. We introduce u = g(x, y) as in Case I and write
u as ((0). We assert that q(0) changes sign in the interval [0, 2jr]. If A and
C are both 0, then B 0 and q(0) = B sin 0 cos 0, so that the assertion follows.
If, for example, A 0, then we can write qT(0) as in (12-241); then T(0) = A
and when0 - Cot-(-B'2A), q(0) has the sign of -A. Hence, (0) changes
sign. Let T(0 i) m1 > 0, (02)= - m2 < 0. Hence, in rectangular coordinates,
Ax"2 + Bxy + Cy2 = m1(x2 + y2) for x= rcos1, y rsin1
The same reasoning as for Case I allows us to conclude that
f(x, y) > f(0, 0) + ~(2 + rf) for x r cos 01, y = r sin 01
provided that 0 K + y2 K 2 Similarly
fxy) f0,0) - ~(x + y2) for x = r cos 2, y -rsin02
provided that 0 < x2 + K - 2. Hence, f can have neither a local maximum
nor a local minimum at (0, 0).
PROBLEMS
1. For each of the following sets determine if the set is closed or not, and in any case
deteri ine the boundary of the set. For (a),..., (d) consider the sets as contained
in B'.
(a) The set of rational numbers of the form 1 /n, n a positive integer.
(b) The set of rational number consisting of 0 and of the nmners of the form 1 n,
n an integer.
(c) The set of all rational numbers with absolute values at most 1.
(d) fThe set of all irrational numbers with absolute value at most 1.
(e) The set of points in the plane with both x and y coordinates rational numbers.
(f) The set of points in the plane satisfying the inequalities: -1 < y sin x,
0<x K2.
g) The set of all points in the plane satisfying the two inequalities:
-1 K y(1 + xi)- 2, 0 K x < 9.
i(h) The set of points in the plane satisfying the inequalities: - 1 < sin (1 x),
f2. (a) By the conmplienit of a set E in R" we mean the set of all points of R not
in E. Show: if E is open, then the complement of E is closed; if E is closed,
then the complement of E is open.
(b) Show that the complement of the union of two sets is the intersection of
their complements.
