Solution to Problem 82-5: A maximum, minimum problem
Citation for published version (APA):
Lossers, O. P. (1983). Solution to Problem 82-5: A maximum, minimum problem. SIAM Review, 25(1), 106-.
https://doi.org/10.1137/1025015
DOI:
10.1137/1025015
Document status and date:
Published: 01/01/1983
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106
PROBLEMS AND SOLUTIONS[4]
G.N. WATSON,
A Treatiseonthe Theoryof
BesselFunctions,Cambridge UniversityPress,
Cambridge, 1958.[5] W.N. BAILEY,
Someinfinite
integrals involvingBesselfunctions, Proc.
LondonMath.Soc., (2)
40(1936),
pp.37-48.
[6] Y. L. LUKE,
Integralsof
Besse!
Functions,McGraw-Hill,New
York,1962.[7]
S.OKUI,
Complete elliptic integrals resultingfrom
infinite
integralsof
BesselFunctionsI, II,
J.Res. Nat.
Bur.
Stand.,78B(1974),
pp.113-135; 79B(1975),
pp.137-170.A
Maximum,
Minimum
Problem
Problem 82-5,
by T. SEIIGUCnI
(University of
Arkansas).
In
Euclidean
space
E3,
with
origin
at
O
and
coordinates
(Xl,
x2,x3),
let
0-be
a
ray
in
the
closed
first
orthant, and
a;
be
the angles between the positive
xi
axes
and
O2,
1,
2,
3.
Determine
the
maximum
and
minimum
values
of
al+
a2+
a3.Solution
by
O.
P.
LOSSERS
(Eindhoven
University of
Technology,
Eindhoven,
the
Netherlands).
Without loss
of generality
we assume
(xl,
x2,x3)
to
be
on
the
unit
sphere
in
the
closed
first octant,
denoted
by
l
x+x+x=l,
S:
/I.X
>=
0,
x2
>--
0,
x3
>--
0.
The
function to
be
considered
then
reads
F(Xl,
x2,x3)
arccos
X--
arccos
X--
arccos
x3.
(i) At
the
boundary
points of
S,
we
have
F(xl,
X2,X3)
71"since
arccos 0
/arccos
x
/arccos
/1
x
(ii) In
the
interior
of
S,
we
find
stationary points for
F
in
those points where grad
F
X grad (Xl
+
x
+
x]-
1),
i.e.,
x/1
x
x2/1
x2
x3/1
x].
Squaring,
we
obtain
x-x-x4+x=0,
etc.
(X
X2)(X
--
X2)(1
X-
X92.)=
0,
etc.
In
this
product
for the
interior
points
only
the
first
factor
can
be
zero,
so
X X X
