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

222_Functions of Several Variables_[Ch. 6
2005. Derive approximate formulas (accurate to second -order
terms in a and P) for the expressions
if |a| and |p| are small compared with unity.
2006*. Using Taylor's formulas up to second-order terms,
approximate
a) 1/T03; ^O98; b) (0.95) 2 - 01 .
2007. z is an implicit function of x and y defined by the
9
equation z 2xz + y = 0, which takes on the value z= 1 for x= 1
and y=l. Write several terms of the expansion of the function
z in increasing powers of the differences x\ and y 1.
Sec. 13. The Extremum of a Function of Several Variables
1. Definition of an extremum of a function. We say that a function
f(x,y) has a maximum (minimum) f (a, b) at the point P (a, b), if for all
points P' (x, y) different from P in a sufficiently small neighbourhood of P
is fulfilled.
the inequality /(a, b) > f(x, y) [or, accordingly, /(a, b) < f (x t y)]
The generic term for maximum and minimum of a function is extremum.
In similar fashion we define the extremum of a function of three or more
variables.
2. Necessary conditions for an extremum. The points at which a differentiate
function f (x, y) may attain an extremum (so-called stationary points)
are found by solving the following system of equations:
t'x (x. 0)-0, f' t/ (x t y)-Q
(1)
(necessary conditions for an extremum). System (I) is equivalent to a single
equation, df(x, #) 0. In the general case, at the point of the extremum
P (a, b), the function f (x, y), or df (a, ft) = 0, or df (a, b) does not exist.
3. Sufficient conditions for an extremum. Let P (a, b) be a stationary
point of the function f(x, y), that is, df (a, &)- 0. Then: a) if d*f (a t b) <
for dx z + dy*>Q t then /(a, b) is the maximum of the function f(x, //); b) if
d z
f(a, ft)>0 for d* 2
-}- di/ 2 > 0, then /(a, b) is the minimum of the function
/(* 0); c ) if d 2
/(a, ft) changes sign, then f (a, b) is not an extremum of /(v, //).
The foregoing conditions are equivalent to the following: let f[ (a, b)----
= f'y (a, ft) -0 and A=fxx (a, ft), B~f xy (a, ft), C = /^(ci, ft). We form the
Then: I) if A > 0, then the function has an extremum at the point
P(a, ft), namely a maximum, if A < (or C < 0), and a minimum, if A >
(or C>0); 2) if A < 0, then there is no extremum at P (a t ft); 3) if A==0.
then the question of an extremum of the function at P (a, ft) remains open
(which is to say, it requires further investigation).
4. The case of a function of many variables. For a function of three or
more variables, the necessary conditions for the existence of an extremum