In mathematics, and especially in the application of mathematics in a number of areas, problems and tasks are encountered in which extremes of function are required, with the variables being bound up with some additional conditions. Extremes of this kind are called conditional extremes. [2]. In this paper, we present the procedure for determining a conditional extremity of a function with two variables, we will present that the point is finding the curve in the space obtained as a cross section on two surfaces and by using free software we will have visualization of the conditional extremum. 

Key words: Conditioned extreme, function, point, visualization

Atanasova-Pacemska, Tatjana

Jusufi Zenku, Teuta

Trifunov, Zoran

UGD Academic Repository

IMO - ISTRAŽIVANJE MATEMATIČKOG OBRAZOVANJA”,  ISSN  2303-4890 (print), ISSN 1986-518X (on line),   DOI: 10.7251/IMO1801033T http://www.imvibl.org/dmbl/meso/imo/imo2.htm     Vol. X (2018), Broj 18, Original research article    Redakcija: Šefket  ARSLANAGIĆ (Sarajevo), asefket@pmf.unsa.ba  Branislav Boričić (Beograd), boricic@ekof.bg.ac.rs  Bernadin  IBRAHIMPAŠIĆ (Bihać), bernadin@bih.net.ba  Slagjana JAKIMOVIK (Skopje), slagjana.jakimovik@gmail.com Duško JOJIĆ (Banja Luka ), ducci68@teol.net Zagorka LOZANOV-CRVENKOVIĆ (Novi Sad), zagorka.lozanov-crvenkovic@dmi.uns.ac.rs  Zoran LUČIĆ (Beograd), zlucic@matf.bg.ac.rs Violeta MARINOVA The University of Veliko Turnovo,  Bugarska, violetmar@abv.bg Mirela MRĐA (Sombor), mirelamrdja@gmail.com  Zlatan MAGAJNA (Ljubljana), Zlatan.Magajna@pef.uni-lj.si. Dubravka MUJICA (Beograd), dmijuca@fgm.edu.rs Daniel A: ROMANO (Banja Luka), daniel.a.romano@hotmail.com       Glavni i odgovorni urednik: DANIEL A. ROMANO e-mail: daniel.a.romano@hotmail.com           Uputstvo autorima: Rukopisi, sa obaveznim sažetkom na Engleskom jeziku i numeracijom Mathematics Education Subject Classification 97-XX, treba da budu pripremljeni na računaru u Wordu ili TeX-u (u nekoj od njegovih verzija), u PDF formatu, A4 veličina, i u elektronskoj formi proslijeđena jednom od članova Redakciji. Formule treba numerisati u oblim zagradama (); Literaturu citirati brojevima, u uglastim zagradama [], i sabrati na kraju članka. Priloge slati na adresu: IMO, Naučno društvo matematičara Banja Luka , Mladena Stojanovića 2, 78000 Banja Luka , Bosna i Hercegovina.    IMO - ISTRAŽIVANJE MATEMATIČKOG OBRAZOVANJA  se referiše / indeksira se u      Math Educt Database (EU - Njemačka),       Index Copernicus International (Poljska),       Реферативный журнал – Математика (Rusija),      Ulrichsweb / Global Serial Direcory (USA) i      EBSCO (USA), ISSN (p) 2303-4890, ISSN (o) 1986–518X                              ISTRAŽIVANJE MATEMATIČKOG OBRAZOVANJA DOI: 10.7251/IMO1801033T                                                           http://www.imvibl.org/dmbl/meso/imo/imo2.htm    Vol. X (2018), Broj 18, 33–40 Original research article  Conditioned Extreme, Determination and Visualization Zoran Trifunov, Teuta Jusufi-Zenku         University “Mother Teresa”, Skopje, Republic of Macedonia,  zoran.trifunov@unt.edu.mk, teuta.zenku@unt.edu.mk,  Tatjana Atanasova-Pachemska  3University “Goce Delcev”, Stip, Republic of Macedonia, tatjana.pacemska@ugd.edu.mk  Abstract. In mathematics, and especially in the application of mathematics in a number of areas, problems and tasks are encountered in which extremes of function are required, with the variables being bound up with some additional conditions. Extremes of this kind are called conditional extremes. [2]. In this paper, we present the procedure for determining a conditional extremity of a function with two variables, we will present that the point is finding the curve in the space obtained as a cross section on two surfaces and by using free software we will have visualization of the conditional extremum. Key words: Conditioned extreme, function, point, visualization.  I. Introduction The determination of the extreme value of a function under a given condition is called the procedure for determining the Conditional extreme. We will describe the procedure for determining conditional extremes of a function with two variables below. We will specify which conditions are necessary to be assigned so that we can determine a conditional extremity of a function in space. But we will also show that the point in which the conditional extremal is located is a point in space [5], and not a point in the plane xOy. (as described in some literature sources [1], [2], [3], [4]) Let the function z = f (x, y) be given, where x and y are two independent variables that represent a surface in space D. And let a certain condition (x,y) = 0 be given. The equation (x,y) = 0 in the plane xOy is a curve, but represented in a space it is a surface C normal on the xOy plane. C surface intersects the surface D on a curve L. The solution to this problem is to find the point M (x0, y0) in the equation (x,y) = 0 in which the function z = f (x, y) has an extreme value, [1], [2], [3], [4]. But if we visualize this problem in space then we will have to find the point M (x0, y0, z0) in the curve L in which the curve L will have an extremity [5]. The point M (x0, y0, z0) is called the point of the conditional extremity of the function z = f (x, y) and in the point M (x0, y0, z0) the function z = f (x, y) does not have an extreme value.  II. Methods for determining conditional extremes How to determine the extreme value of a function under a given condition? IMO, X (2018), Broj 18                                                                            Z. Trifunov, T. Jusufi-Zenku  and  T. Atanasova - Pachemska                                                                                                                               34  Determining the extreme value of the contour (curve L) is reduced to the determination of the extreme value of the function z = f (x, y) which is achieved under given conditions for (x,y) = 0. There are two methods for finding a conditional extreme. 1. Method of direct elimination The extreme value of the function z = f (x, y) under the condition (x,y) = 0 can be found such that y is expressed from the equation (x,y) = 0 in the explicit form y = y (x) and is replaced in the function z = f (x, y). In this way a function with one variable z = f (x, y (x)) is obtained, and the task is reduced to determining the extreme value of a function of an independent variable. The derivation of the function after the variable x at the point of the conditional extremity should be 0dzdx . Then the procedure for determining the extremity of a function with one variable is used. The second procedure for determining a conditional extremal is used when the elimination of y is complicated and it is desirable to remain on the functions z = f (x, y) and (x,y) = 0. 2. Lagrange method for multipliers How is the Lagrange function (= Lagrangian expression) defined for a real function of two variables given by the equation z = f (x, y) in the search of a conditional extremal given by the conditional equation (x,y) = 0? First is calculate the derivative of z, z zdz dx dyx y    By dividing with dx and we get dz z z dydx x y dx    and because 0dzdx  is getting                                                           0z z dyx y dx                                                                              (1) The derivative dydx is obtained by differentiating the conditional equation (x,y)=0 by x, i.e.                                                           0dyx y dx                                                                          (2) From what we get dyy dx x      dy xdxy  If dydx is replaced in (1) is obtained IMO, X (2018), Broj 18                                                                            Z. Trifunov, T. Jusufi-Zenku  and  T. Atanasova - Pachemska                                                                                                                               35  0z z xx yy     or z zx y y x          If we mark the proportions with -, we get zzyxx y       where  is a constant called the Lagrange multiplier. The values of x, y,   for which the function has an extremum will be found by solving the system obtained from the three equations zxx  , zyy , ( , ) 0x y   i.e. system equations are                                                                         00( , ) 0zx xzy yx y                                                                   (3) System solutions give the values for x, y and  for which the function can have an extremal value at the given condition. With additional tests or on the basis of the nature of the task, it is concluded whether that point is a minimum or a maximum. III. Calculation of conditional extremes with the Lagrange method What is the procedure for determining the conditional extremes with Lagrange method of undetermined multipliers ( , , ) ( , ) ( , )F x y f x y x y      From the functions z=f(x,y) and the condition (x,y) = 0 by introducing a multiplier  a new function is formed, so called the Lagrange function with the equation ( , , ) ( , ) ( , )F x y f x y x y      It determines the values of x, y and  that satisfy the system equations (3) where the left sides are partial derivatives of the function ( , , )F x y   IMO, X (2018), Broj 18                                                                            Z. Trifunov, T. Jusufi-Zenku  and  T. Atanasova - Pachemska                                                                                                                               36  ' 0' 0' 0xyFFF  2 '' 2 '' '' 2( , , ) 2k k k xx xy yyd F x y F dx F dxdy F dy     For obtained values (xk, yk, k) the values of the expression are determined, 2 '' 2 '' '' 2( , , ) 2k k k xx xy yyd F x y F dx F dxdy F dy     Provided that ( , ) 0x y  . If - 2 0d F  , then the function z = f(x,y) has a minimum in the point Mk(xk, yk, zk) of the curve L, - 2 0d F  , then the function z = f(x,y) has a maximum in the point Mk(xk, yk, zk) of the curve L.  Example: Determine the extreme value of the function ( , ) 2z f x y x y    on condition 2 2 5x y   i.е 2 2( , ) 5x y x y    . Visualization in GeoGebra ( , ) 2z f x y x y   is the surface D 2 2( , ) 5x y x y     is the surface C L is the curve in the space that is the intersection of D and C.   Solution: With the Lagrange method of multipliers, we form the function IMO, X (2018), Broj 18                                                                            Z. Trifunov, T. Jusufi-Zenku  and  T. Atanasova - Pachemska                                                                                                                               37  2 2( , , ) 2 ( 5)F x y x y x y       . Partial derivatives are determined, equated to 0, and the system equations are obtained: ' 0' 0' 0xyFFF  т.е. 2 21 2 02 2 05 0xyx y      We get a solution to the system 112    and  1 11, 2x y   i.е. point M1(1,2). If we replace it in the equation ( , ) 2z f x y x y   , then we get 1 5z  . We got a stationary point М(1,2,5) from L in which the function can have a conditioned extremity.   We get a solution to the system 212   и  2 21, 2x y     i.е. point N1(-1,-2). If we replace it in the equation ( , ) 2z f x y x y   , then we get 2 5z   . We got another stationary point N(-1,-2,-5) from L in which the function can have a conditioned extremity.  IMO, X (2018), Broj 18                                                                            Z. Trifunov, T. Jusufi-Zenku  and  T. Atanasova - Pachemska                                                                                                                               38  The second derivatives of the function 2 2( , , ) 2 ( 5)F x y x y x y        are: '' 2xxF   '' 2yyF   '' 0xyF   For 112    and 1 11, 2x y  , we get '' 1xxF    '' 1yyF    '' 0xyF  . The value of the expression 2 '' 2 '' '' 2 2 22 1 1 0xx xy yyd F F dx F dxdy F dy dx dy       . Follows the function has a conditioned maximum in the point М(1,2,5)=Mmax  For 212    and 2 21, 2x y    , we get '' 1xxF   '' 1yyF   '' 0xyF  . The value of the expression 2 '' 2 '' '' 2 2 22 1 1 0xx xy yyd F F dx F dxdy F dy dx dy      . Follows the function has a conditioned minimum in the point N(-1,-2,-5)=Nmin.  IMO, X (2018), Broj 18                                                                            Z. Trifunov, T. Jusufi-Zenku  and  T. Atanasova - Pachemska                                                                                                                               39  Visualization of the example in GeoGebra. The functions are entered through the input bar in GeoGebra in the drawing pad and the 3D graph. The intersection of the objects is determined the curve L is obtained for which the extreme value is determined. After the algebraic determination of the extreme values of the value bar, the points are entered and visually represented in 3D. Checkboxes specify what will be displayed.    IV Conclusion A conditioned extreme is a small problem of mathematics that has been applied in other areas. To understand the laws that apply to solving these problems, and to have greater application, it is necessary to visualize them. Visualization should use software that visually poses a problem. In this visualization problem we introduced the point of the conditional extremum and showed that it is a point that lies on the curve in the space L, not the curve in xOy.  Reference [1]  B. M. Pipevski. Mathematical Analysis, II. Faculty of Electrical Engineering, University of Skopje, Skopje 2005 [Боро М. Пиперевски . Математичка анализа II.  Електротехнички факултет, Скопје 2004.] [2]  E. Atanasova, S. Geogievska. Mathematics, II. St. Kiril and Metodie University, Skopje 2002. [Елена Атанасова, Слободанка Георгиевска.  Математика II. Универзитет „Св. Кирил и Методиј“ Скопје, 2002.] [3]  Hass Thomas Weir. Thomas’ Calculus. Pearson Addison-Wesley, 2005. [4]  Howard Anton, Irl Bivens, Stephen Davis. Calculus early transcendental. John Wiley & Sons, INC, 2009 IMO, X (2018), Broj 18                                                                            Z. Trifunov, T. Jusufi-Zenku  and  T. Atanasova - Pachemska                                                                                                                               40  [5]  N.Maličić and M. Maličić. Elements of Higher mathematics, II. Academska misao, Belgrade 2005. [Нада Миличич, Милош Миличич . Елементи на виша математика (II дел). Академска мисао, Београд, 2005.] [6]   https://www.geogebra.org/m/YEg7m9bp 

Conditioned Extreme, Determination and Visualization

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Conditioned Extreme, Determination and Visualization

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