Note – Second derivative test
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If
U
⊆
R
n
is a
closed and bounded set
and
f
:
R
n
→
R
is continuous on
U
, then
f
has a global maximum and minimum on
U
. To ﬁnd these using the second derivative
test, proceed as follows when
U
has a smooth boundary:
•
Determine the critical points of
f
by solving
∇
f
= 0.
◦
Discard critical points which are not contained in
U
.
•
Determine
f
(
a
) for each critical point
a
∈
U
.
•
Use the second derivative test to determine which values
f
(
a
) are local
minima, maxima on
U
, and where the test fails.
•
On the boundary
∂U
of
U
, use the preceding steps to determine all
local extremes.
•
Compare all the values found to ﬁnd the global minima and maxima.
The second derivative test (the third step) can fail even though
f
(
a
) is a local maxi
mum or minimum. For example, if
f
(
x,y,z
) =
x
4
+
y
4
+
z
4
, it is clear that
f
(0
,
0
,
0) = 0
is a local minimum of
f
, whereas the Hessian matrix is the allzero matrix, and the
second derivative test fails. The
boundary
∂U
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 Spring '07
 Enright
 Math, Derivative

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