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Saddle Points and Inflection Points

​
f(x,y) =
2
y
-
2
x
​
f(x,y) and g(x)=
{
2x
ifx<0
x
2
ifx≥0
f(x,g(x))=
{
3
2
x
ifx<0
-
3
2
x
4
ifx≥0
Theorem: Let
f
be a function with continuous second partial derivatives in a open set
U
in the plane and let
(a,b)
be a saddle point in
U
. Then there exists a continuous function
y=g(x)
with
g(a)=b
for which the projection on the
xz
plane of the intersection of the surface
z=f(x,y)
and the cylindrical surface
y=g(x)
has a inflection point at
x=a
.
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