Introduction

In mathematics ,the maximum and minimum of a function,known collectively as extrema are the largest and smallest value that 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 first mathematicians to propose a general technique (called adequality) for finding maxima and minima.To locate extreme values is the basic objective of optimization.

PIERRE DE FERMAT ; 17 August 1601 12 January 1665) was a French lawyer at the Parlement of Toulouse, France, and a mathematician who is given credit for early developments that led to innitesimal calculus, including his technique of adequality. In particular, he is recognized 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 dierential calculus, then unknown, and his research into number theory. He made notable contributions to analytic geometry, probability, and optics. He is best known for Fermat's Last Theorem, which he described in a note at the margin of a copy of Diophantus' Arithmetica

PART 1

Mathematical Optimization

Mathematical optimization (alternatively, optimization or mathematical programming) is the selection of a best element (with regard to some criteria) from some set of available alternatives.

In the simplest case, an optimization problem consists of maximizing or minimizing a real function by systematically choosing input values from within an allowed set and computing the value of the function. The generalization of optimization theory and techniques to other formulations comprises a large area of applied mathematics. More generally, optimization includes finding "best available" values of some objective function given a defined domain (or a set of constraints), including a variety of different types of objective functions and different types of domains.The general process of finding an optimum is called programming

Optimization can be represented in the following ways :

Given: a function f:S from some set s to the real numbers

An optimization problem can be described using a function f:S that is a function from the space S to a real number.

The space S is some subset of a euclidean space n for some N.

We call it the objective function if we are looking for a maximum. Each point in S is called a potential or feasible solution for the objective function.

We call it cost function if we are looking for a minimum.

Sought : an element x0 in A such that f(x0) f(x) or f(x0) f(x for all x in A .

When the function is on minimization ,it will be F(x0) f(x) for all x in A .

When the function is on maximization,it will be f(x0) f(x) for all x in A.

Global And Local Maximum And Minimum

Part 2

A ) Encik Shah has been involved in a sheep farming business for several years and he supplies meats and milk products to the communities. He wishes to build a rectangular sheep pen with two parallel partitions using 200 metres fence.Find the dimension of the rectangle that will maximize the total area of the pen.Hence state the maximum area of the pen.

x

x

Sheep Pen

y

2y + 2x = 200 m

2y= 200-2X

Y=100-X

Maximum area = 100(50)-(50)

=2500 m

XY is equation number 2

100 2X = 0

X= 50

Formula for calculating Area is xy

Substitute X=50 into 2

Since the area is maximum,

Substitute 1 into 2

Y=100 x is equation number 1

b) Reza is helping En Shah to make a box without the top.The box is made by cutting away four square from the corners of a 30 cm square piece of cardboard as shown in figure below. And bending up the resulting cardboad to form the walls of the box.

Find the largest possible volume of the box.

Maximum volume,

This will be equation number 1

Since it is maximum,we will take h=15 and substitute into 1

Since the volume is maximum,

12h 240h + 900 = 0

(h 15) ( h 5) = 0

part 3

A market research company finds that traffic in a local mall over the course of a day could be estimated by function where P,is the number of people going to the mall,and t ,is the time, in hours, after the mall opens. The mall opens at 9:30 a.m.

i) Sketching the graph

ii) Peak hours and number of people.

Based on the graph, the peak hours will be 3: 30 pm, 6 hours after the mall Is open. The number of people is 3600.

iii) Estimate the number of people in the mall at 7:30 pm.

substitute t=10 into the function.

At 7 : 30 pm, it is 10 hours after the mall I open.

t = 10

iv) The time when number of people reach 2570.

+1800

The time,

Further Exploration