Appendix A
Review of Mathematical Results
A.1
Local Minimum and Maximum for Multi-Variable
Functions
Recall that for a single-variable function,
f
(
x
), local extrema (local maxima or minima)
can occur only at
critical points
. For single-variable functions, the critical points are the
points
x
0
where
f
0
(
x
0
) = 0 or where
f
(
x
0
) is undefined. In multivariable calculus, where
there is more than one independent variable, say
f
(
x, y
), we have
The critical points of
f
(
x, y
) are those points, (
x
0
, y
0
), where either:
1.
∂f
∂x
=
∂f
∂y
= 0,
or
2.
f
(
x
0
, y
0
) is undefined.
A local extrema can occur only at a critical point.
Note that another way of expressing condition 1, above, is to say that the gradient
of
f
,
∇
f
, must be zero.
Now consider the critical points where
f
(
x
0
, y
0
) are defined (i.e. we are in condition
1, not 2).
Then how can we tell if it’s a local minimum, maximum, or saddle point?
Recall that for a single-variable function where
x
0
is a point where
f
0
(
x
0
) = 0, we can
use the second-derivative test: If
f
00
(
x
0
)
>
0 then the function is concave up and
x
0
is
where a local minimum is; if
f
00
(
x
0
)
<
0 then the function is concave down and
x
0
is
where a local maximum is. The equivalent test for multivariable functions is to look at
the
Hessian
of the function
f
(
x, y
), which is a matrix of second partials:
Hf
(
x, y
) =
∂
2
f
∂x
2
∂
2
f
∂x∂y
∂
2
f
∂y∂x
∂
2
f
∂y
2
.
(A.1)
If the critical point, (
x
0
, y
0
), satisfying criterion 1 is where a local minimum is located,
then the Hessian evaluated at (
x
0
, y
0
) is
positive definite
, i.e. all the eigenvalues are
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