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