Math 16B – Spring 2011 – Supplementary Notes 2
SecondDerivative Test
To understand what is behind the secondderivative test for functions of two variables, we shall
start by looking at the simplest nontrivial example, that of a polynomial of degree 2. First the test
will be stated.
Let the function
f
(
x,y
) have a critical point at (
a,b
). The secondderivative test involves the
function
D
f
(
x,y
) =
±
∂
2
f
∂x
2
²±
∂
2
f
∂y
2
²

±
∂
2
f
∂x∂y
²
2
,
and it applies when
D
f
(
a,b
)
6
= 0. Note that if
D
f
(
a,b
)
>
0 then
∂
2
f
∂x
2
(
a,b
) and
∂
2
f
∂y
2
(
a,b
) must have
the same sign. The test distinguishes three cases:
(I) If
D
f
(
a,b
)
>
0 and
∂
2
f
∂x
2
(
a,b
)
>
0 (equivalently
∂
2
f
∂y
2
(
a,b
)
>
0), then (
a,b
) is a relative minimum
of
f
.
(II) If
D
f
(
a,b
)
>
0 and
∂
2
f
∂x
2
(
a,b
)
<
0 (equivalently
∂
2
f
∂y
2
(
a,b
)
<
0), then (
a,b
) is a relative maximum
of
f
.
(III) If
D
f
(
a,b
)
<
0 then (
a,b
) is a saddle point of
f
(neither a relative maximum nor a relative
minimum).
Now we look at the simple example
f
(
x,y
) =
αx
2
+2
βxy
+
γy
2
, where
α,β,γ
are constants, not
all 0. This quadratic polynomial has a critical point at (0