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

• knowLittle
In summary, a local maximum is a point on a graph where the value of the function is greater than all the values of the function in its immediate vicinity, while a local minimum is a point on a graph where the value of the function is smaller than all the values of the function in its immediate vicinity. To find these points, the derivative of the function is taken and critical points, where the derivative equals zero, are identified. These points provide information about the behavior of the function, such as the highest and lowest points and any potential turning points. A saddle point is a point of inflection on the graph where the function has a critical point but is neither a local maximum nor a local minimum. The second derivative test can be used to
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?

Is this correct?

knowLittle said:

## 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\ ##?

You are right fxy= -4.

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

knowLittle said:
You are right fxy= -4.

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

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.

## 1. What is the definition of a local maximum and minimum?

A local maximum is a point on a graph where the value of the function is greater than all the values of the function in its immediate vicinity. Similarly, a local minimum is a point on a graph where the value of the function is smaller than all the values of the function in its immediate vicinity.

## 2. How do you find the local maxima and minima of a function?

To find the local maxima and minima of a function, you need to take the derivative of the function and find the points where the derivative equals zero. These points are known as critical points and can be used to determine the local maxima and minima of the function.

## 3. What is the significance of finding the local maxima and minima of a function?

The local maxima and minima of a function can provide valuable information about the behavior of the function. They can indicate the highest and lowest points of the function, as well as any potential turning points. This information can be useful in understanding the overall shape and behavior of the function.

## 4. What is a saddle point and how is it different from local maxima and minima?

A saddle point is a point on a graph where the function has a critical point but is neither a local maximum nor a local minimum. At a saddle point, the function has a slope in one direction and a slope in the opposite direction, making it a point of inflection rather than an extremum.

## 5. How do you determine if a critical point is a local maximum, minimum, or saddle point?

To determine the type of critical point, you can use the second derivative test. If the second derivative at the critical point is positive, it is a local minimum. If the second derivative is negative, it is a local maximum. If the second derivative is zero, further analysis is needed to determine if it is a saddle point.

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