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Copyright 2007 Samuel Dagan
dagan@post.tau.ac.il
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Sufficient conditions
Classification of stationary points, two variables
The origin of the figure is at (0,0)
©
Samuel Dagan
Sufficient conditions
A stationary
point of f = f(x, y)
is marked.
fxx depends on the
direction of the drawn Δx.
A rotation of the coordinate
system by an angle
Δφ
about the origin yields a different value.
Indeed fx'x' = cos²Δφ
G(t) , where
t = tanΔφ
and G(t) = fyyt² + 2fxyt + fxx .
The sufficient condition for a
minimum at a
stationary point states that
fx'x' > 0 for
any Δφ,
or G(t) > 0
for any t,
as drawn next.
Although the quadratic function
G( t)
differs in
shape and location in each case,
the stated
above condition must hold, as
illustrated next.
The requirement that
G( t)
does not have
roots is equivalent to
fxxfyy
− (fxy
)²
> 0. For a
minimum
one requires also that fxx(or fyy) > 0.
The sufficient condition for a
maximum at a
stationary point states that
fx'x' < 0 for
any Δφ,
or G(t) < 0
for any t,
as drawn next.
This is equivalent to the
requirements of fxxfyy
− (fxy
)²
> 0 and of fxx(or fyy) < 0.
In the case of a saddle point,
the sign of fx'x' should alternate
between different
regions of Δφ .
This is equivalent to
G( t)
with alternating signs,
obtained by two roots of G( t) ,
as drawn next.
The sufficient condition for a
saddle point becomes fxxfyy
− (fxy
)²
< 0 for any fxx and fyy .
fxxfyy
− (fxy
)²
= 0 remains
unresolved, and needs further investigation for each
particular case.
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