Page 206 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 8] APPLICATIONS OF PARTIAL DERIVATIVES 197

2
The expression B AC also has the property of invariance with respect to plane rotations
x ¼ x cos y sin
x
y
y ¼ x sin þ y cos
y
x
It has been discovered that with the identifications A ¼ f xx ; B ¼ f xy ; C ¼ f yy ,we have the partial deri-
2
vative form f xy f xx f yy that characterizes relative extrema.
The demonstration of invariance of this form can be found in analytic geometric books. However, if
you would like to put the problem in the context of the second partial derivative, observe that
@x @y
x
f x ¼ f x þ f y ¼ f x cos þ f y sin
x
@ x @ x
x
@x @y
f y ¼ f x þ f y ¼ f x sin þ f y cos
y
@ y y @ y y
Then using the chain rule to compute the second partial derivatives and proceeding by straightforward
but tedious calculation one shows that
2
2
f xy ¼ f xx f yy ¼ f x f x f y :
xy
xx
y
x
y
yy
The following equivalences are a consequence of this invariant form (independently of direction in the
tangent plane at P 0 ):
2
f xy f xx f yy < 0 and f xx f yy > 0 ð1Þ
2
f xy f xx f yy > 0 and f xx f yy < 0 ð2Þ
The key relation is (1)because in order that this equivalence hold, both f x f y must have the same sign.
We can look to the one variable case (make the same argument for each coordinate direction) and conclude
that there is a relative minimum at P 0 if both partial derivatives are positive and a relative maximum if both
are negative. We can make this argument for any pair of coordinate directions because of the invariance
under rotation that was established.
If (2) holds, then the point is called a saddle point.Ifthe quadratic form is zero, no information results.
Observe that this situation is analogous to the one variable extreme value theory in which the nature of
f at x, and with f ðxÞ¼ 0, is undecided if f ðxÞ¼ 0.
00
0
3
3
8.22. Find the relative maxima and minima of f ðx; yÞ¼ x þ y 3x 12y þ 20.
2
2
f x ¼ 3x 3 ¼ 0 when x ¼ 1; f y ¼ 3y 12 ¼ 0 when y ¼ 2. Then critical points are Pð1; 2Þ,
Qð 1; 2Þ; Rð1; 2Þ; Sð 1; 2Þ.
2
f xx ¼ 6x; f yy ¼ 6y; f xy ¼ 0. Then ¼ f xx f yy f xy ¼ 36xy.
At Pð1; 2Þ; > 0 and f xx (or f yy Þ > 0; hence P is a relative minimum point.
At Qð 1; 2Þ; < 0 and Q is neither a relative maximum or minimum point.
At Rð1; 2Þ; < 0 and R is neither a relative maximum or minimum point.
At Sð 1; 2Þ; > 0 and f xx (or f yy Þ < 0so S is a relative maximum point.
Thus, the relative minimum value of f ðx; yÞ occurring at P is 2, while the relative maximum value
occurring at S is 38. Points Q and R are saddle points.
8.23. A rectangular box, open at the top, is to have a volume of 32 cubic feet. What must be the
dimensions so that the total surface is a minimum?
If x, y and z are the edges (see Fig. 8-7), then
Volume of box ¼ V ¼ xyz ¼ 32
ð1Þ
Surface area of box ¼ S ¼ xy þ 2yz þ 2xz
ð2Þ
or, since z ¼ 32=xy from (1),
64 64
x y Fig. 8-7
S ¼ xy þ þ