Problems in Mathematical Analysis.pdf - pwp.net.ipl.pt

Sec. 13]_The Extremum of a Function of Several Variables_223
are similar to conditions (1), while the sufficient conditions are analogous to
the conditions a), b), and c) 3.
Example 1. Test the following function for an extremum:
or
Solution. Find the partial derivatives and form a system of equations (1):
r ** ( + /*_5-0,
\ xy 2 = 0.
Solving the system we get four stationary points:
P,(l,2); P t (2, 1); P,(-l,-2); P 4 (_2,-1).
Let us find tiie second derivatives
d 2 z c d 2 z r d*z c
dx 2
y
dy 2
a- = 6.v, 3 T- = 6r/, T-2 = 6x
dxdy
and form the discriminant A=^/4C B 2 for each stationary point.
1) For the pomt P t : A
= (g} =6. B = (fL\ =12, C=(g) =
\dx 2
Jp l \dxdyjp, \dy 2
J p,
2 = 6, A^=4C = 36 144 < 0. Thus, there is no extremum at the point P,.
2) For the point P 2 : 4
--12, B^6, C-12; A = 144 36 > 0, /I > 0. At P 2
the function has a minimum. This minimum is equal to the value of the
function for A -2, y~\'
) 3012^28.
3) For the point P : 9 ^-6,
i no extremum.
fi--- 12, C^ 6; A = 36 144 < 0. There
At
4) For the point P :
4 ^-
the point P 4 the function
12, B = 6, C= 12; A = 144
has a maximum equal to 2ma x
36 > 0, A < 0.
^ 6-f-30-{-
4- 12 ---28
5*. Conditional extremum. In the simplest case, the conditional extremum
of a function /(A, //) is a maximum or minimum of this function which is
attained on the condition that its arguments are related by the equation
(A-, i/) = we form the so-called Lagra