INTRODUCTIONIn mathematics, the maximum and minimum of a function, known collectively as extrema are largest and smallest value that the function takes at a point either within a given neighbourhood (local or relative extremum) or on the function domain in its entirely (global or absolute extremum). Pierre de Fermat was one of the rst mathematicians to propose the general techni!ue (called ad e!uality) for nding maxima and minima point. "o locate extreme values is the basic ob#ective ofoptimi$ation Pierre de Fermat (French% &p#'() d*f')ma+, -. /ugust -01- 2 -3 4anuary -005) was a French lawyer at the Parlement of "oulouse, France, and a mathematician whois given credit for early developments that led to innitesimal calculus, including histechni!ue of ad e!uality. In particular, he is recogni$ed for his discovery of an original method of nding the greatest and the smallest ordinates of curved lines, which is analogous to that of the di6erential calculus, then unknown, and his research into number theory. 7e made notable contributions to analytic geometry, probability, and optics. 7e is best known for Fermat8s 9ast "heorem, which he described in a note at the margin of a copy of :iophantus8 /rithmetica.-PART Ia) :escribei. Mathematical optimization: In mathematics, computer science, operations research, mathematical optimi$ation (alternatively, optimi$ation or mathematical programming) is theselection of a best element (with regard to some criteria) from some set of available alternatives. In the simplest case, an optimi$ation problem consists of maximi$ing or minimi$ing a real function by systematically choosing input values from within an allowed set and computing the value of the function. "he generali$ation of optimi$ation theory and techni!ues to other formulations comprises a large area of applied mathematics. ;ore generally, optimi$ation includes nding b# Method$ o% &ndin' the maim!m and minim!m (al!e o% a )!adratic %!nctionMethod * o% +: I% the )!adratic i$ in the %orm , - a. / b / cDecide 0hether ,o!1re 'oin' to &nd the maim!m (al!e or minim!m (al!e. It8s either one or the other, you8re not going to nd both. "he maximum or minimum value of a !uadratic function occurs at its vertex.For y ? ax3 @ bx @ c,2c 3 b.45a# gives the yAvalue (or the value of the function) at its vertex.If the value of a is positive, you8re going to get the minimum value because as such the parabola opens upwards (the vertex is the lowest the graph can get)5If the value of a is negative, you8re going to nd the maximum value because as such the parabola opens downward (the vertex is the highest point the graph can get)"he value of a can8t be $ero, otherwise we wouldn8t be dealing with a !uadratic function, would weB0Method . o% +: I% the )!adratic i$ in the %orm , - a23h#. / 6For , - a23h#. / 676 i$ the (al!e o% the %!nction at it$ (erte. 6 gives us the maximum or minimum value of the !uadratic accordingly as a is negative or positive respectively..Method + o% +: U$in' di8erentiation 0hen the )!adratic i$ in the %orm , - a9. / b / cDi8erentiate , 0ith re$pect to . dyCdx ? 3ax @ bDetermine the di8erentiation point (al!e$ in term$ o% d,4d. Dince dyCdx is the gradient function of a curve, the gradient of a curve at any given point can be Efound. "hus, the maximumCminimum value can be found by setting these values e!ual to 1 and nd the corresponding values. dyCdx ? 1. 3ax@b ? 1, x ? AbC3a:!b$tit!te thi$ (al!e o%into , to 'et the minim!m4maim!m point.i3Thin6 MapFPART II-1PART III--FURT;-, Frank 9auren 7itchcock also formulated transportation problems as linear programs and gave a solution very similar to the later Dimplex method,&3+ 7itchcock had died in -F5. and the Mobel pri$e is not awarded posthumously.:uring -F>0A-F>., Neorge O. :ant$ig independently developed general linear programming formulation to use for planning problems in PD /ir Force. In -F>., :ant$ig also invented the simplex method that for the rst time eGciently tackled the linear programming problem in most cases. Hhen :ant$ig arranged meeting with 4ohn von Meumann to discuss his Dimplex method, Meumann immediately con#ectured the theory of duality by reali$ing that the problem he had been working in game theory was e!uivalent. :ant$ig provided formal proof in an unpublished report the eld came in -FE> when Marendra Karmarkar introduced a new interiorApoint method for solving linearAprogramming problems.-5U:5(01) @ 05(01) ? 0011>5(-31) @ 05(>1) ? E111>5(=1) @ 05(=1) ? ==11;aximum% E111;inimum% ==11