Find All local maxima and minima and all saddle points of the function

[knowLittle]

Homework Statement

## f\left( x,y\right) =x^{2}-4xy+6x-8y+2y^{2}+10 ##

## f_{x}=2x-4y+6=0 ##
## f_{y}=-4x-8+4y=0 ##

## f_{y}=-4\left( x-y+2\right) ##
-2=x-y, then solving fx and using this equality

## f_{x}=0=2\left( x-2y+3\right) =0 ##

2(-2 -y+3)=0
2(1-y)=0
y=1, then pluggin it to values

-2=x-y
-2=x-1
-1=x, So critical points at (-1,1)

fxx(-1,1)fyy(-1,1)-0^2=8, which is greater than zero and fxx is too. Therefore, there is a global minima at (-1,1)

There is no saddle point or global maxima?

Homework Equations

The Attempt at a Solution

Is this correct?

 

[LCKurtz]

Homework Statement

## f\left( x,y\right) =x^{2}-4xy+6x-8y+2y^{2}+10 ##

## f_{x}=2x-4y+6=0 ##
## f_{y}=-4x-8+4y=0 ##

## f_{y}=-4\left( x-y+2\right) ##
-2=x-y, then solving fx and using this equality

## f_{x}=0=2\left( x-2y+3\right) =0 ##

2(-2 -y+3)=0
2(1-y)=0
y=1, then pluggin it to values

-2=x-y
-2=x-1
-1=x, So critical points at (-1,1)

fxx(-1,1)fyy(-1,1)-0^2=8, ...[/b]

Ok until right there. How did you get ##f_{xy}=0\ ##?

 

[knowLittle]

You are right fxy= -4.

So,
fxx(-1,1)fyy(-1,1)-(-4^2)=
fxx=2;
fyy=4;
8-16=-8
Answer:
So, (-1,1) is a saddle point and there is not enough information(Domain) to find local extrema.

Is this answer correct?

 

[LCKurtz]

You are right fxy= -4.

So,
fxx(-1,1)fyy(-1,1)-(-4^2)=
fxx=2;
fyy=4;
8-16=-8
Answer:
So, (-1,1) is a saddle point and there is not enough information(Domain) to find local extrema.

Is this answer correct?

Yes, it is a saddle point. But I wouldn't say "there is not enough information(Domain) to find local extrema". I would say there are no relative extrema.