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Classify extreme point of $f(x,y)$
I'm currently studying the first course of multivariable calculus and I have a question.
Suppose $f'_x (a,b) = f'_y (a,b) = 0$, i.e., we have an extreme point at $(x,y)=(a,b)$. To identify the type of this point I would like to use the same method as we use in one variable calculus, but twice. My solution would then look like the following:
if $f''_{xx} (a,b) < 0$ and $f''_{yy} (a,b) < 0$ we have a local maximum point.
if $f''_{xx} (a,b) > 0$ and $f''_{yy}(a,b) > 0$ we have a local minimum point.
if $f''_{xx} (a,b)$ and $f''_{yy} (a,b)$ have different signs, we have a saddle point.
The book suggests that you compute $(f''_{xy} (a,b))^2 f''_{xx} (a,b) \cdot f''_{yy} (a,b)$, etc.
Doesn't my method work? If it doesn't, why not? Thanks.
Your "method" is used to determine the max/min in any case.
The second derivative test is $f_{xx}f_{yy}  f^2_{xy} <0$
and
$f_{xx}f_{yy}  f^2_{xy} >0 $
$$f_{xx}f_{yy}  f^2_{xy} <0 \quad \text{,is a saddle point}$$
$$f_{x

Your "method" is used to determine the max/min in any case.
The second derivative test is $f_{xx}f_{yy}  f^2_{xy} <0$
and
$f_{xx}f_{yy}  f^2_{xy} >0 $
$$f_{xx}f_{yy}  f^2_{xy} <0 \quad \text{,is a saddle point}$$
$$f_{xx}f_{yy}  f^2_{xy} >0 \quad \text{,is a min or a max}$$
If $ f_{xx}< 0$ and $f_{yy} <0$ then you have a max
If $ f_{xx}> 0$ and $f_{yy} >0$ then you have a min
EDIT: By Example Suppose $f(x,y)=x^2y^2$ , then
$f_{xx}(0,0) = 2 <0$ and
$f_{yy}(0,0) = 2 <0$
and your method works since we have a max at $(0,0)$
Suppose now however that I have the function $f(x,y) = x^4y^2xy$, then your first derivative tests are satisfied at $(0,0)$ but
$f_{xx}(0,0) = 0$ and
$f_{yy}(0,0) = 2 <0$
Here we have that $f_{xx} = 0$ and your method fails to describe what happens at this point. Might be a poor example, but long story short is that your method would be ignoring points.
20180312 21:17:42