760
C H A P T E R
15
DIFFERENTIATION IN SEVERAL VARIABLES
(ET CHAPTER 14)
(
x
β
−
1
)
α
−
1
=
x
(
β
−
1
)(
α
−
1
)
=
x
β
α
·
α
β
=
x
By properties of inverse functions, their graphs are symmetric with respect to the line
y
=
x
. Therefore the area between
one graph and the
x
-axis is equal to the area between the other graph and the
y
-axis, over the intervals 0
≤
x
≤
R
and 0
≤
y
≤
R
, respectively. In particular, the integral
R
0
x
β
−
1
dx
can be thought of as the area between the graph
of
y
=
x
α
−
1
and the
y
-axis, from
y
=
0 to
y
=
R
. So, recalling that the area of the squares is
R
2
, and assuming that
α
≥
2
≥
β
, we have the following inequality (see figure, and note that the first integral gives the area of the lower-right
part of the square as well as that extra spike at the top of the square, while the second integral gives the area of the rest of
the square, as per our discussion above):
R
2
≤
R
0
x
α
−
1
dx
+
R
0
x
β
−
1
dx
x
y
=
x
b
−
1
y
=
x
a
−
1
y
R
R
1
1
(1, 1)
This proves (1).
47.
The following problem was proposed by Fermat as a challenge to the Italian scientist Evangelista Torricelli
(1608–1647), a student of Galileo and inventor of the barometer. Given three points
A
=
(
a
1
,
a
2
)
,
B
=
(
b
1
,
b
2
)
, and
C
=
(
c
1
,
c
2
)
in the plane, find the point
P
=
(
x
,
y
)
that minimizes the sum of the distances
f
(
x
,
y
)
=
AP
+
B P
+
C P
(a)
Write out
f
(
x
,
y
)
as a function of
x
and
y
, and show that
f
(
x
,
y
)
is differentiable except at the points
A
,
B
,
C
.
(b)
Define the unit vectors
e
=
−→
AP
−→
AP
,
f
=
−→
B P
−→
B P
,
g
=
−→
C P
−→
C P
Show that the condition
∇
f
=
0 is equivalent to
e
+
f
+
g
=
0
3
Prove that Eq. (3) holds if and only if the mutual angles between the unit vectors are all 120
◦
.
(c)
Define the Fermat point to be the point
P
such that angles between the segments
AP
,
B P
,
C P
are all 120
◦
. Conclude
that the
Fermat point
solves the minimization problem (Figure 23).
(d)
Show that the Fermat point does not exist if one of the angles in
ABC
is
≥
120
◦
. Where does the minimum occur
in this case?
Hint:
The minimum must occur at a point where
f
(
x
,
y
)
is not differentiable.