Page 196 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 8] APPLICATIONS OF PARTIAL DERIVATIVES 187

b b u 2 u 2 b
ð ð ð ð ð
f ðx; Þ d dx
ð Þ d ¼ f ðx; Þ dx d ¼ ð18Þ
a a u 1 u 1 a
The result is known as interchange of the order of integration or integration under the integral sign. (See
Problem 8.18.)

MAXIMA AND MINIMA
In Chapter 4 we briefly examined relative extrema for functions of one variable. The general idea
was that for points of the graph of y ¼ gðxÞ that were locally highest or lowest, the condition g ðxÞ¼ 0
0
was necessary. Such points P 0 ðx 0 Þ were called critical points. (See Fig. 8-3a,b.) The condition g ðxÞ¼ 0
0
was useful in searching for relative maxima and minima but it was not decisive. (See Fig. 8-3(c).)

Fig. 8-3 Fig. 8-4

To determine the exact nature of the function at a critical point P 0 , g ðx 0 Þ had to be examined.
00
> 0 counterclockwise rotation (rel. min.)
g ðx 0 Þ < 0 implied a clockwise rotation (rel. max)
00
¼ 0 need for further investigation.
This section describes the necessary and sufficient conditions for relative extrema of functions of two
variables. Geometrically we think of surfaces, S, represented by z ¼ f ðx; yÞ. Ifata point P 0 ðx 0 ; y 0 Þ
then f x ðx; y 0 Þ¼ 0, means that the curve of intersection of S and the plane y ¼ y 0 has a tangent parallel to
the x-axis. Similarly f y ðx 0 ; y 0 Þ¼ 0 indicates that the curve of intersection of S and the cross section
x ¼ x 0 has a tangent parallel the y-axis. (See Problem 8.20.)
Thus

f x ðx; y 0 Þ¼ 0; f y ðx 0 ; yÞ¼ 0

are necessary conditions for a relative extrema of z ¼ f ðx; yÞ at P 0 ; however, they are not sufficient
because there are directions associated with a rotation through 3608 that have not been examined. Of
course, no differentiation between relative maxima and relative minima has been made. (See Fig. 8-4.)
A very special form, f xy f x f y invariant under plane rotation, and capable of characterizing the
2
roots of a quadratic equation, Ax þ 2Bx þ C ¼ 0, allows us to form sufficient conditions for
relative extrema. (See Problem 8.21.)