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Maximum and Minimum Condition of function of two (or more) variables 

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Maximum and Minimum Condition of function of two (or more) variables 

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A local maximum and minimum of the function is a point inside the domain when the value is greater than the other point.

To decide the Local Maximum or Minimum of capacity we should to know experienced in geometry, mechanics, material science, and different fields, and was one of the persuading factors in the improvement of the math

In Local Maximum and Minimum Condition we will find out about The Second Derivatives.

The Second Derivative is utilized to decide whether a basic point is a relative Maximum or relative Minimum.

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Maxim and minimum condition of function of two (or more) variables?

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Maxim and minimum condition of function of two (or more) variables?

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A local maximum and minimum of the function is a point inside the domain when the value ia greater than the other point.

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-Maximum and Minimum Condition of Function of Two (or more) Variable-

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-Maximum and Minimum Condition of Function of Two (or more) Variable-

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The maximum and minimum of a function is very useful in geometry, physics, mechanics, and other fields. Maximum and minimum of function of two (or more) variable is simply talking about the partial derivatives of a function to find its maximum and minimum. Every function that has two or more variables has a local maximum at point (a,b) if f(x,y) ≤ f(a,b) for all points (x,y) in some disk with center (a,b). Then, the value of f(a,b) is called a local maximum value of f. According to  Calculus III e-book, we can know the theorem for finding the maximums and minimums:
Theorem : If f  is differentiable and has a local maximum or minimum at (a,b),
then fx(a,b) = 0 and fy(a,b) = 0.
For more explanation, you can kindly check this link: https://math.oregonstate.edu/home/programs/undergrad/CalculusQuestStudyGuides/vcalc/min_max/min_max.html
Thank you.

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In calculus, the maxiuma and minimum condition, are the largest and smallest value of the function,  are the condition when the value is either maxium or minimum. It can be in spesific range or in the entire domain of the function. I just watch the video material from youtube. https://www.youtube.com/watch?v=V239IIoGVT8 

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In calculus, the maxiuma and minimum condition, are the largest and smallest value of the function,  are the condition when the value is either maxium or minimum. It can be in spesific range or in the entire domain of the function. I just watch the video material from youtube. https://www.youtube.com/watch?v=V239IIoGVT8 

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Hope it can help your understanding

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The thing that I know from maximum and minimum is the way to find it.
Second derivative of a function will determine the maximum and minimum.
I suggest you guys to open this link http://hyperphysics.phy-astr.gsu.edu/hbase/Math/maxmin.html. It helps you to understand by stated the example of maximum and minimum calculation.

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Understanding Maximum and Minimum Condition of a Function (Part 1)

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Understanding Maximum and Minimum Condition of a Function (Part 1)

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Hey guys! Since everyone has shared the ideas of finding the minimum and maximum conditions of two variable, I will try to explain it very quickly and in a fun way! Also you can check out more about the detail in the link down below :
https://www.mathsisfun.com/calculus/maxima-minima.html

Finding Maxima and Minima using Derivatives

Where is a function at a high or low point? Calculus can help!

A maximum is a high point and a minimum is a low point. In a smoothly changing function a maximum or minimum is always where the function flattens out  (except for a saddle point).

Where does it flatten out?  Where the slope is zero.

Where is the slope zero?  The Derivative tells us!

Second Derivative Test

When a function's slope is zero at x, and the second derivative at x is:

less than 0, it is a local maximum

greater than 0, it is a local minimum

equal to 0, then the test fails (there may be other ways of finding out though)

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Maximum and minimum conditions of functions of two or more variables

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Maximum and minimum conditions of functions of two or more variables

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[Kistian Soleman Kuli]

To find the local maximum and
local minimum in two or more variables we have to solve the partial derivative.
At two variable and centered at disk (a,b), then the  local maximum at (a,b) if f(x,y) ≤ f(a,b) and  local minimum  f(a,b) ≤  f(x,y) for all point (x,y)
with the equation
D(a,b) = fxx(a,b)fyy(a,b) – { fxy(a,b) }^2

If the critical point can not determine maximum and minimum, then it is a saddle point (showed on the picture).

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Maximum and Minimum Conditions of Function in Two or More Variable

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Maximum and Minimum Conditions of Function in Two or More Variable

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based on book Calculus 3 tell me that use second derivatives to determine either the maximum or minimum of some f (function).
We can use the Formula of second derivatives where  fx(a,b) = 0 and fy(a,b) = 0

so,  D(a,b) = fxx(a,b)fyy(a,b) – { fxy(a,b) }^2

I believe all of you know it well if Local maximum  at (a,b) if f (x,y) ≤ f (a,b) for all points

(x,y)  with center (a,b). Also for the feature materials can we get form the class and book.

We need to learn this condition to understand the function either max or min.

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Maximum and Minimum Condition of Function in Two or More Variable     (Nur Fatikhah)

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Maximum and Minimum Condition of Function in Two or More Variable     (Nur Fatikhah)

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Based on Calculus 3 E-book
To determining the maximum or minimum of function we must to know encountered in geometry, mechanics, physics, and other fields, and was one of the motivating factors in the development of the calculus
This is the same with the materials of Calculus 1.

In Maximum and Minimum Condition we will learn about The Second Derivative.

The Second Derivative is employed to determine if a critical point is a relative maximum or relative minimum.

f''(x_c)>0, then x_c is a relative minimum.
f''(x_c)<0, then x_c is a relative maximum .
The notions of Critical points and the second derivative is over to function of two variables (Let z=f(x,y))

For The Second Derivative Test for Function of Two Variable.

Let (x_c, y_c) be a critical point.
We have the following cases:

If D>0 and f_xx(x_c,y_c)<0, then f(x,y) has a relative maximum at (x_c,y_c).

If D>0 and f_xx(x_c,y_c)>0, then f(x,y) has a relative minimum at (x_c,y_c).

If D<0, then f(x,y) has a saddle point at (x_c,y_c).

If D=0, the second derivative test is inconclusive.

Maximum and Minimum in Bounded Region
- Has three types of points that can potentially to be maximum or minimum

1. Relative extrema in the interior of the square.

2. Relative extrema on the boundary of the square.

3. Corner Points.

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Maxima and Minima of two variable function in hindi

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Maximum and Minimum Condition of function in two or more variable-Dilla Ayu Wardani

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Maximum and Minimum Condition of function in two or more variable-Dilla Ayu Wardani

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According to E-book Calculus III that I read, local maximum and minimum in two or more variables can be determined by finding second (mixed partial) derivative  test. Local maximum in two or more variables (a,b) is defined if  f(x,y) ≤ f(a,b) for all points (x,y) in some center disk (a,b). In addition, if the function is differentiable and has local maxima or minima in (a,b), so the fx (a,b) and  fy (a,b) equal to zero.

Second derivative test is used to determine whether the function is local maxima, local minima or the saddle point.

Here is the rule for second derivative test:

If we have fxx , fxy , fyx , and fyy are continuous in a disk with center (a,b) and fx(a,b) = 0 and fy(a,b) = 0. Let D = D(a,b) = fxx(a,b)fyy(a,b) – { fxy(a,b) }

(i)                   If D > 0 and fxx(a,b) > 0, then f(a,b) is a local minimum.

(ii)                 (ii) If D > 0 and fxx(a,b) < 0, then f(a,b) is a local maximum.

(iii)                (iii) If D < 0, then f(a,b) is nor in a local minimum or a local maximum. (It is a saddle point.)

(iv)               (iv) If D = 0, then the test is "indeterminate": f(a,b) could be a local maximum or a local

minima or neither.

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"maximum and minimum conditions" of functions of two (or more) variables?

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"maximum and minimum conditions" of functions of two (or more) variables?

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I agree with Krishna. this is similar with calculus 1. but differences with calculus 1 and 3 is variables. if I not mistaken in calculus 1 we just learn in Maximum and minimum condition in one variable and then how if the variable more than one. According the research that I can find maximum and minimum in two variablels or more in can be happen. for example f(x) = x^3 - 4x^2 + 5x + 2 when f'(x) = 0 . it can see in the link that I'am given. the concept same with one variables but in one variable just oce differential in the two or more variable have twice differential.
This is that I get from that link. if I'm mistaken understand. let me know. someone can add comment in my opinion. it is my plesure if you add the comment for me. Thank you.

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A thorough and highly accessible resource for analysts in a broadrange of social sciences. Optimization: Foundations and Applications presents a series ofapproaches to the challenges faced by analysts who must find thebest way to accomplish particular objectives, usually with theadded complication of constraints on the available choices.Award-winning educator Ronald E.

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Krishna Dharma [Max and Min Conditions of Multivariable functions}

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Krishna Dharma [Max and Min Conditions of Multivariable functions}

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It is basically similar to what was learnt in Calculus 1, however this time we will be dealing with two or more variables. The concept is still the same which is finding the relative max or min points by finding out their partial differentials fx and fy. A critical point of a multivariable function is a point where the partial derivatives of first order of this function are equal to zero. We can use it to determine whether the point is a local maximum or a local minimum.

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Faridah Tampubolon - Maximum and Minimum Conditions of Functions of two (or more) Variables

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Faridah Tampubolon - Maximum and Minimum Conditions of Functions of two (or more) Variables

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This is using second derivative test. Suppose that the second partial derivatives of F are continuous, with center (a,b) and suppose that fx(a,b)=0 and fy(a,b)=0.
The formula is D= D(a,b)= fxx(a,b)fyy(a,b)- fxy(a,b).
This of maximum o minimum point can be defined in to 4 conditions such as;
1. if D>0 and fxx(a,b) >0, then f(a,b) is a local minimum
2. if D>0, fxx(a,b)<0, then f(a,b) is a local maximum.
3. if D<0, then f(a,b) is not a local max/min
4. If D = 0, then the test is "indeterminate": f(a,b) could be a local max/min or neither.

Here I attached the example of "Find local max, local min, saddle points"

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Maximum and Minimum Condition of Function of Two or More Variables

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Maximum and Minimum Condition of Function of Two or More Variables

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So, basically to determine whether a critical point was a local maximum or minimum using the second derivatives.

I've understand that the value f(a,b) is a local max.  if f (x,y) ≤ f (a,b) for all points (x,y).

And there re four conditions that we must aware if we want to determine max. or min.

(i)  If D > 0 and fxx (a,b) > 0, then f (a,b) is a local minimum.

(ii) If D > 0 and fxx (a,b) < 0, then f (a,b) is a local maximum.

(iii) If D < 0, then f (a,b) is not a local minimum or a local maximum. (It is a saddle point.)

(iv) If D = 0, then the test is "indeterminate": means it could be neither local max. nor local min.

In this case fxx and fyy will be the same because both of them are multiplication.

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Learn what local maxima/minima look like for multivariable function.

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Maximum and Minimum Condition of Function of Two or More Variable [Anisa Pitriani K]

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Maximum and Minimum Condition of Function of Two or More Variable [Anisa Pitriani K]

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Based on what I read on calulus 3 book, I got a definition:
"A function of two variables f(x,y) has a local maximum at (a,b) if f(x,y) ≤ f(a,b) for all points (x,y) in some disk with center (a,b). The value f(a,b) is called a local maximum value of f."

To determined the maximum and minimum value of two or more Variable, we can use a second (mixed partial) derivative, where :
fx(a,b) = 0 and fy(a,b) = 0
so,  D(a,b) = fxx(a,b)fyy(a,b) – { fxy(a,b) }^2

1. If D > 0 and fxx(a,b) > 0, then f(a,b) is a local minimum.
2. If D > 0 and fxx(a,b) < 0, then f(a,b) is a local maximum.
3. If D < 0, then f(a,b) is a saddle point.)
4. If D = 0, then f(a,b) could be a local maximum or a local minimum or neither.

For the question example, click this video

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Local Maximum and Minimum Values -Function of Two Variables

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Maximum and Minimum Condition of Function of Two Variable [Maulidina Lubis]

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Maximum and Minimum Condition of Function of Two Variable [Maulidina Lubis]

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In my understanding, a function of two variable has maximum at (a,b) if f(a,b)  ≥  f(x,y). The maximum value of the function is f(a,b) and this occurs at (a,b). Global maximum is the highest point. However, f(a,b) ≤  f(x,y) (it is show that the function has minimum.

Finding maximum and minimum function. If the function define on open interval, and f(a,b) is a local extreme of f, then f'(a,b)=0

Please remind me if I have a misunderstanding

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Maximum and Minimum Conditions of Function of Two or More Variable-fiella Pramysilia Citra

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Maximum and Minimum Conditions of Function of Two or More Variable-fiella Pramysilia Citra

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The term of maximum,  means the function's value is in the peak. While minimum condition shows that the function has the value in the bottom.

In order to determine whether a function's value is maximum or minimum, firstly,  we suppose that:
The second partial derivatives continuous on a disk centered in f(a,b), then fx(a,b) and fy(a,b) are equal to 0.
Therefore,
D(a,b)=fxx(a,b)fyy(a,b)-fxy(a,b).
Results :
1. Minimum, whenever D>0 and fxx(a,b)>0.
2. Maximum, whenever D>0 and fxx(a,b)<0.
3. Saddle point,  where D<0 and fxx(a,b) is neither maximum nor minimum.

For the better understanding, just check it out!  :)
https://youtu.be/Hm5QnuDjNmY

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Maximum and Minimum Condition of Function of Two Variable [Angelia Tarigan]

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Maximum and Minimum Condition of Function of Two Variable [Angelia Tarigan]

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Here I attach my opinion about maximum and minimum point of more than 1 variables based on the theory that I got from the E-book and a video from Youtube.
Click here to see the video  :)

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Maximum and Minimum points of more than 1 variables

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"Maximum and Minimum Conditions" of Functions of Two (or more) Variables - Shafira R. Khoirun Nisa

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"Maximum and Minimum Conditions" of Functions of Two (or more) Variables - Shafira R. Khoirun Nisa

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In a single variable function, we use derivatives to find maximum and minimum. Similarly, in multivariate function, to find the maximum and minimum we also use derivatives, but it is a partial derivative.

A multivariate function centered at (a,b) has a local maximum at (a,b) if f(x,y) ≤ f(a,b) for all points (x,y) and the local minimum value is if the f(x,y) ≥ f(a,b) for all points (x,y).

There is also a theorem that tells about where we should look for maximums and minimums.  If f is differentiable and has a local maximum or minimum at (a,b), then fx (a,b) = 0 and fy (a,b) = 0.

To determine the local maximum and minimum of a multivariate function, we also can use a second (mixed partial) derivative test. Here is the link for more explanation about second derivatives test:

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Determine the critical points and locate any relative minima, maxima and saddle points of function f defined by . Find the first partial derivatives f. (x,y) = 4x + 2y - 6 (x,y) = 2x + 4y The critical points satisfy the equations f. . We now need to find the second order partial derivatives f(x,y) = 4 (x,y) = 4 (x,y) = 2 = 12 Since D is positive and f Determine the critical points and locate any relative minima, maxima and saddle points of function f defined by Determine the critical points and locate any relative minima, maxima and saddle points of function f defined by Determine the critical points of the functions below and find out whether each point corresponds to a relative minimum, maximum, saddle point or no conclusion can be made. - 2 xy - 8x - 9y

f(x , y) = 2x² + 2xy + 2y² - 6x

Solution to Example 1: x and f
f y
f x
x(x,y) = 0 and f
4x + 2y - 6 = 0
2x + 4y = 0
xx(x,y), f
f yy(x,y) and f
f xy(x,y).
f

y(x,y) = 0 simultaneously. Hence
D = f yThe above system of equations has one solution at the point (2,-1) xx yy xyWe now need to find D defined above. xx(2,-1) f yy(2,-1) - f xy ²(2,-1) = ( 4 )( 4 ) - 2
xx(2,-1) is also positive, according to the above theorem function f has a local minimum at (2,-1).
The 3-Dimensional graph of function f given above shows that f has a local minimum at the point (2,-1,f(2,-1)) = (2,-1,-6).
²

Example 2

f(x , y) = 2x² - 4xy + y⁴ + 2
.
A 3-Dimensional graph of function f shows that f has two local minima at (-1,-1,1) and (1,1,1) and one saddle point at (0,0,2).

Find the first partial derivatives fDetermine the critical points by solving the equations fThe first equation gives x = y. Substitute x by y in the equation - 4x + 4yWe now use a table to study the signs of D and f Solution to Example 2:
x and f y.
f x(x,y) = 4x - 4y
f y(x,y) = - 4x + 4y ³ x(x,y) = 0 and f y(x,y) = 0 simultaneously. Hence.

4x - 4y = 0
- 4x + 4y ³ = 0 to obtain. ³ = 0

- 4y + 4y ³ = 0
Factor and solve for y.
4y(-1 + y ²) = 0
y = 0 , y = 1 and y = -1
We now use the equation x = y to find the critical points.
(0 , 0) , (1 , 1) and (-1 , -1)
We now determine the second order partial derivatives.
f xx(x,y) = 4
f yy(x,y) = 12y ²
f xy(x,y) = -4
xx(a,b) and use the above theorem to decide on whether a given critical point is a saddle point, relative maximum or minimum.

Example 3

f(x , y) = - x⁴ - y⁴ + 4xy
.

A 3-Dimensional graph of function f shows that f has two local maxima at (-1,-1,2) and (1,1,2) and a saddle point at (0,0,0).

1. f(x , y) = x

First partial derivatives fWe now solve the equations fThe first equation gives y = xWe now use the equation y = xThe table below shows the signs of D and f Solution to Example 3:
x and f y are given by.
f x(x,y) = - 4x ³ + 4y
f y(x,y) = - 4y ³ + 4x y(x,y) = 0 and f x(x,y) = 0 to find the critical points..

- 4x ³ + 4y = 0
- 4y ³. Combined with the second equation, we obtain. ³ + 4x = 0

- 4(x ³) ³ + 4x = 0
Which may be written as .
x(x ⁴ - 1)(x ⁴ + 1) = 0
Which has the solutions.
x = 0 , -1 and 1.
³ to find the critical points.
(0 , 0) , (1 , 1) and (-1 , -1)
We now determine the second order partial derivatives.
f xx(x,y) = -12x ²
f yy(x,y) = -12y ²
f xy(x,y) = 4
xx(a,b). Then the above theorem is used to decide on what type of critical points it is.

2. f(x , y) = x
Exercises
² + 3 y ² ³ - 12 x + y ³ + 3 y
²
1. relative maximum at (1,1) and (-1,-1) and a saddle point at (0,0)
2. relative maximum at (2,-3), relative minimum at (2,1), saddle points at (-2,-3) and (-2,1).

Answers to the Above Exercises

More Links on Partial Derivatives and Multivariable Functions
Multivariable Functions Home Page

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Maxima and Minima of Functions of Two Variables

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Determine the critical points and locate any relative minima, maxima and saddle points of function f defined by . Find the first partial derivatives f. (x,y) = 4x + 2y - 6 (x,y) = 2x + 4y The critical points satisfy the equations f. .

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"Maximum and Minimum Conditions" of Functions of Two (or more) Variables by Gemintang Bening Segara ASri

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"Maximum and Minimum Conditions" of Functions of Two (or more) Variables by Gemintang Bening Segara ASri

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From what I read, I got that;
when there is a factor with two or more variables with continuous second order partial derivatives fxx, fyy, fxy at a critical point (a,b) then:

D (a,b) = fxx(a,b) fyy(a,b) - [fxy(a,b)]^2

- If D > 0, and fxx(a,b) > 0, then f has a relative minimum at (a,b)
- If D > 0, and fxx (a,b) < 0, then f has relative maximum at (a,b)
- If D < 0, then f(x,y) has a saddle point at (a,b)
- If D = 0, the second derivative test is inconclusive.

For better understanding, check this video!

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A description of maxima and minima of multivariable functions, what they look like, and a little bit about how to find them.

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Maximum and Minimum Conditions of two variables by M. Alfin Prayogo

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Maximum and Minimum Conditions of two variables by M. Alfin Prayogo

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A function ,f, of two variables is said to have a relative maximum or minimum at a point (a,b) if there is a disk centered at (a,b) , then f(a,b) ≥  f(x,y) (shows relative maximum), while f(a,b) ≤  f(x,y) (shows relative minimum) for all points (x,y) that lie inside in the disk.

a function ,f, is said to have an absolute maximum or minimum at (a,b) if f(a,b) ≥ f(x,y) (absolute maximum), while f(a,b) ≤ f(x,y) (absolute minimum)

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Max and Min Conditions of function of two/more variables by Agnes Ruth Ahimsa

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Max and Min Conditions of function of two/more variables by Agnes Ruth Ahimsa

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In e-book I've read that the definitions is a function of two variables F(X,Y) has a ,local maximum at (a,b) if f(x,y)  < f(a,b) for all points (x,y) in some disk with center (a,b). The value f(a,b) is called a local maximum of f.

And for the theorem :
If --> F is differentiable and has a local maximum or minimum  s at (a,b)

Then --> Fx(a,b) =0 Fy 9a,b) = 0.

For the question example --> Find any local maximum&minimum of F(x,y) = x^2 - y^2!!

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This is just an example. You might also attach file or website to support your opinion and discussion.

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You can also post more than one comment in here. Use English and appropriate language, okay?

Happy discussing!! :)