This preview shows pages 1–2. Sign up to view the full content.
Study sheet — Stationary Points
Harry Dankowicz
Mechanical Science and Engineering
University of Illinois at UrbanaChampaign
danko@uiuc.edu
Objective:
To investigate the conditions for a local maximum or minimum of a function.
Observation 1
Consider the function
f
(
x, y
)=5
x
−
y
. As the rate of change of
f
with respect to either of
its arguments is constant and nonzero,
f
either increases or decreases under any variations in either of
its arguments and remains constant if and only if
x
and
y
vary along a line in
R
2
with slope 5. Each
such line may be characterized by the intersection with the
y
axis, say
y
∗
, such that
f
(
x, y
)=
−
y
∗
along this entire line. It follows that the value of
f
decreases as
y
∗
increases.
Now restrict attention to those points in
R
2
that live on the unit circle centered on the origin, i.e.,
such that
x
2
+
y
2
=1
.
(1)
We know that
f
is constant along each member of the family of straight lines in
R
2
with slope 5.
Only a subset of these lines intersect the unit circle. Of these, the one with the smallest value of
y
∗
corresponds to the largest value of
f
along the circle. Similarly, the one with the largest value of
y
∗
corresponds to the smallest value of
f
along the circle. Each of these two lines are tangent to the circle
at the corresponding points of intersection. The corresponding points of intersection must therefore lie
on a straight line through the origin with slope
−
1
/
5. It follows that
f
μ
1
√
26
,
−
5
√
26
¶
=
10
√
26
(2)
is the largest value of
f
along the circle and
f
μ
−
1
√
26
,
5
√
26
¶
=
−
10
√
26
(3)
is the smallest value of
f
along the circle.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
 Spring '08
 Weaver

Click to edit the document details