Graphics Reference
In-Depth Information
4.5.5. Example.
To analyze the extrema of the function
(
)
=
22
fxy
,
xy
on the unit disk.
Solution.
Looking for critical points alone will find its minimum at (0,0) but not its
maximum, which lies on the boundary of the unit disk. To find that, make the sub-
stitutions x = cos q and y = sin q and define a function
()
=
2
2
=
()
2
g q
cos
q
sin
q
14
sin
2
q
.
We only need to find the maxima of the function g(q) for qŒ[0,2p]. Now
¢
()
=
=
()
g q
sin
22
q
cos
q
24
sin
q
.
Therefore, g¢(q) = 0 implies that q=k(p/4), for k = 0,1, . . . ,7. The second derivative
test shows that g takes on its maximum values when q=k(p/4), for k = 1,3,5, and 7.
Another common type of extremum problem is finding extrema subject to certain
constraints.
4.5.6. Example.
Maximize the function f(x,y) = xy subject to the condition that
x
2
+ y
2
= 1.
The straighforward way to solve this problem would be to solve for y in the
constraint equation and then substitute this into the formula for f thereby reducing
the problem to a problem about extrema of functions of one variable. Unfortunately,
this is not always feasible since the constraints may be much more complicated.
We therefore want to briefly mention another popular approach to these types of
problems. We begin with two facts that motivate the approach. We shall see the
second, Proposition 4.5.8, again in Chapter 8 when we discuss tangent vectors to
surfaces.
4.5.7. Proposition.
Let
X
Õ
R
n
be an open set and assume that f :
X
Æ
R
and g : [a,b]
Æ
X
are differentiable functions. If
p
=g(c) is an extremum of f along g for some c in
(a,b), then —f(
p
)•g¢(c) = 0.
Proof.
Consider the function g(t) = f(g(t)). Since c is an extremum for g, it follows
that g¢(c) = 0. The result now follows from the chain rule, which says that
¢
()
=—
(
()
)
∑
()
gt
f t
g
g
t
.
4.5.8. Proposition.
Let g :
R
n
Æ
R
be a differentiable function. If g : [a,b] Æ g
-1
(0) is
a differentiable function, then —g•g¢ = 0.
Proof.
Consider the function h(t) = g(g(t)). By hypothesis, h(t) = 0 for all t. There-
fore, h¢(t) = 0 and the chain rule gives the result.
