“TERM PAPER IN“TERM PAPER IN MATHEMATICS”MATHEMATICS”

TOPIC: optimization of maxima andTOPIC: optimization of maxima and minimaminima

SUBMITTED TOSUBMITTED TOMISS.CHEENA GUPTAMISS.CHEENA GUPTALECTURER, LPULECTURER, LPU

SUBMITTED BY:SUBMITTED BY:

Diwakar singhDiwakar singhROLL R246B32ROLL R246B32

SECTION 246SECTION 246Regd. No.10800771 Regd. No.10800771

ACKNOWLEDGMENTMathematics has always been an interesting subject to deal with, and it’s the pillar of engineering. As being an engineering student I feel it very interesting in dealing with the mathematical project.I have been allotted the project on “optimization of maxima and minima”.I feel very thankful to my mathematics teacher Miss.Cheena Gupta whose able guidance helped me to complete my project in time.Secondly I would like to thank my parents, without whom it has not been possible. Thank you Mom and Dad.My honour to the God almighty who gave me wisdom to do everything..Finally my deepest thanks to my friends who stand by me in every turn of life.

CONTENTS

1 Definitions

2 History

3 Graphical representation 4 Finding maxima and minima 5 Examples 6 Functions of more variables

Definitions

A real-valued function f' defined on the real line is said to have a local maximum point at the point x∗, if there exists some ε > 0, such that f(x∗) ≥ f(x) when |x − x∗| < ε. The value of the function at this point is called maximum of the function. ...

On a graph of a function, its local maxima will look like the tops of hills.

Similarly, a function has a local minimum point at x∗, if f(x∗) ≤ f(x) when |x − x∗| < ε. The value of the function at this point is called minimum of the function.

On a graph of a function, its local minima will look like the bottoms of valleys.

A function has a global maximum point at x∗, if f(x∗) ≥ f(x) for all x.

Similarly, a function has a global minimum point at x∗, if f(x∗) ≤ f(x) for all x.

Any global maximum (minimum) point is also a local maximum (minimum) point; however, a local maximum or minimum point need not also be a global maximum or minimum point.

Terminology: The terms local and global are synonymous with relative and absolute respectively. Also extremum is an inclusive term that includes both maximum and minimum: a local extremum is a local or relative maximum or minimum, and a global extremum is a global or absolute maximum or minimum.

Restricted domains: There may be maxima and minima for a function whose domain does not include all real numbers. A real-valued function, whose domain is any set, can have a global maximum and minimum. There may also be local maxima and local minima points, but only at points of the domain set where the concept of neighborhood is defined. A neighborhood plays the role of the set of x such that |x − x∗| < ε. In mathematics, the domain of a function is the set of all input values to the function. ... In mathematics, the real numbers may be described informally as numbers that can be given by an

infinite decimal representation, such as 2. ... In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... This is a glossary of some terms used in the branch of mathematics known as topology. ...

A continuous (real-valued) function on a compact set always takes maximum and minimum values on that set. An important example is a function whose domain is a closed (and bounded) interval of real numbers (see the graph above). The neighborhood requirement precludes a local maximum or minimum at an endpoint of an interval. However, an endpoint may still be a global maximum or minimum. Thus it is not always true, for finite domains, that a global maximum (minimum) must also be a local maximum (minimum). In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ... The term interval is used in the following contexts: cricket mathematics music time This is a disambiguation page &#8212; a navigational aid which lists other pages that might otherwise share the same title. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... This is a glossary of some terms used in the branch of mathematics known as topology. ...

Terminology: The term optimum can replace either one of the terms maximum or minimum, depending on the context. Some optimization problems (see next paragraph) search for a global maximum value while others search for a global minimum value.

Solving an Ancient Optimization

problem

Euler said: “Nothing takes place in the world whose meaning is not that of some maximum or minimum.” (Tikhomirov, 1990).

The study of optimization problems began approximately twenty-five centuries ago and they are important in many areas of life today. These problems provide excellent examples for the application of mathematical concepts but relatively few students have the opportunity to solve “real world” optimization problems. These problems have traditionally been relegated to the calculus curriculum due to the level of analytic skills required for solution.

Strategies that could be used for solving optimization problem are sometimes neglected in favor of analytic methods. Many of these strategies can be considered using the TI-NspireTM. The TI-NspireTM allows students to investigate a problem through linked geometric, algebraic, and numeric explorations and facilitates multiple representations.

Heron of Alexandria was interested in all forms of measurement as well as optics and mechanics. He considered this problem in the 1st century A.D.:A and B are two given points on the same side of line L. Find a point D on L such that the sum of the distances from A to D and from D to B is a minimum Tikhomirov, 1990)

Problems of this type are still found in calculus texts. This is a modification of one example: Town A is 14 kilometers from a river and town B is nine kilometers from that same river. Town B is on the same side of the river as town A. The river follows a straight path between locations O and C as shown below. The distance from town A to town B is 13 kilometers. A pumping station is to be built along the river to supply water to both towns. Where should the pumping station be built so that the sum of the distances from the pumping station to the two towns is a minimum?

Many calculus teachers find that even those students who are able to successfully solve optimization problems seldom enjoy them – many are put off by the tedious calculations needed to solve these problems. Simpler geometric methods of solution may be neglected and Lagrange’s approach is the only one found in many classrooms. “The methods I set forth require neither constructions nor geometric nor mechanical considerations. They require only algebraic operations subject to a systematic and uniform course.” (Lagrange quoted in Tikhomirov, 1990)

Problem 1: An Investigative Method

As shown in Figure 1, information on the problem is provided on a model of the problem is constructed (Figure 2). drag D and simply estimate the best location for the pumping station (D).

find the distance from point O to point C (Figure 3). When they drag point D on , the distances from A to D, from B to D, and the sum of the two distances are displayed (Figure 4).

Figure 2

Figure 3 Figure 4

Figure 1

Students are asked to estimate the best location for the pumping station in terms of the distance from point O to point C.

Problem 2: Heron’s Method

The problem posed by Heron concerned the physical behavior of light rays. According to historians, the law of reflection of light was not new - it was known to Euclid, Aristotle, and probably also Plato. Boyer and Merzbach (1989) note that Heron gave a simple geometric argument in his work on reflection (Catoptrics) to prove that the angle of incidence equaled the angle of reflections. Kline (1959) stated that Heron used the “shortest possible path” characteristic of light waves and congruent triangles in his proof. Kline also noted that the shortest path requirement is found in many other applications.

(Figure 5) provides directions with a different scale and a hint (Figure 6). Point B’ is the reflection of point B about the line segment OC.

A line segment from point A to point B’ is drawn and the intersection point of segments AB’ and OC is found and labeled (Figure 7). The distance from point O to point D is measured (Figure 8).

Figure 5 Figure 6

Problem 3: A Geometric Method

In problem 3, (Figures 9 and 10). Similar triangles may be used to solve the problem.

Figure 7 Figure 8

Figure 9 Figure 10

GRAPHICAL REPRESENTATION

Local and global maxima and minima for cos(3πx)/x, 0.1≤x≤1.1In mathematics, maxima and minima, known collectively as extrema, are the largest value (maximum) or smallest value (minimum), that a function takes in a point either within a given neighbourhood (local extremum) or on the function domain in its entirety (global extremum). Image File history File links Extrema_example. ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A... In mathematics, the domain of a function is the set of all input values to the function. ...

FINDING MAXIMA AND MINIMA

Finding global maxima and minima is the goal of optimization. If a function is continuous on a closed interval, then by the extreme value theorem global maxima and minima exist. Furthermore, a global maximum (or minimum) either must be a local maximum (or minimum) in the interior of the domain, or must lie on the boundary of the domain. So a method of finding a global maximum (or minimum) is to look at all the local maxima (or minima) in the interior, and also look at the maxima (or minima) of the points on the boundary; and take the biggest (or smallest) one. In mathematics, the term optimization, or mathematical programming, refers to the study of problems in which one seeks to minimize or maximize a real function by systematically choosing the values of real or integer variables from within an allowed set. ... A continuous function in a closed interval has a minimum (blue) and a maximum (red). ...

Local extrema can be found by Fermat's theorem, which states that they must occur at critical points. One can distinguish whether a critical point is a local maximum or local minimum by using the first derivative test or second derivative test. Fermats theorem is a theorem in real analysis, named after Pierre de Fermat. ... In mathematics, a critical point (or critical number) is a point on the domain of a function where: one dimension: the derivative is equal to zero or does not exist: it is points that are either stationary points or non-differentiable points. ... In calculus, a branch of mathematics, the first derivative test determines whether a given critical point of a function is a maximum, a minimum, or neither. ... In calculus, a branch of mathematics, the second derivative test determines whether a given stationary point of a function (where its first derivative is zero) is a maximum, a minimum, or neither. ...

For any function that is defined piecewise, one finds maxima (or minima) by finding the maximum (or minimum) of each piece separately; and then seeing which one is biggest (or smallest). In mathematics, a function f(x) of a real number variable x is defined piecewise, if f(x) is given by different expressions on various intervals. ...

Different strategies may be employed to solve the problem algebraically. Students may find the slopes of segment AB and segment AD (or segment DB’), set the two slopes equal to each other, and solve for x.

Examples

x3+3x2−2x+1−4 ≤ x ≤ 2

The function x2 has a unique global minimum at x = 0. The function x3 has no global or local minima or maxima. Although the

first derivative (3x2) is 0 at x = 0, this is an inflection point. The function x3/3 − x has first derivative x2 − 1 and second derivative

2x. Setting the first derivative to 0 and solving for x gives stationary points at −1 and +1. From the sign of the second derivative we can see that −1 is a local maximum and +1 is a local minimum. Note that this function has no global maximum or minimum.

The function |x| has a global minimum at x = 0 that cannot be found by taking derivatives, because the derivative does not exist at x = 0.

The function cos(x) has infinitely many global maxima at 0, ±2π, ±4π, …, and infinitely many global minima at ±π, ±3π, ….

The function 2 cos(x) − x has infinitely many local maxima and minima, but no global maximum or minimum.

The function cos(3πx)/x with 0.1 ≤ x ≤ 1.1 has a global maximum at x = 0.1 (a boundary), a global minimum near x = 0.3, a local maximum near x = 0.6, and a local minimum near x = 1.0. (See figure at top of page.)

The function x3 + 3x2 − 2x + 1 defined over the closed interval (segment) [−4,2] has two extrema: one local maximum at x = −1−√15⁄3, one local minimum at x = −1+√15⁄3, a global maximum at x = 2 and a global minimum at x = −4. (See figure at right)

Functions of more variables

For functions of more than one variable, similar conditions apply.

The global maximum is the point at the top.For example, in the (enlargeable) figure at the right, the necessary conditions for a local maximum are similar to those of a function with only one variable. The first partial derivatives as to z (the variable to be maximized) are zero at the maximum (the glowing dot on top in the figure). The second partial derivatives are negative. These are only necessary, not sufficient, conditions for a local maximum because of the possibility of a saddle point. For use of these conditions to solve for a maximum, the function z must also be differentiable throughout. The second partial derivative test can help classify the point as a relative maximum or relative minimum.

A Counter example

Counterexample described in the textHowever, for identifying global maxima and minima, there are substantial differences between functions of one and several variables. For example, if a differentiable function f defined on the real line has a single critical point, which is a local minimum, then it is also a global minimum (use the

intermediate value theorem and Rolle's theorem to prove this by reductio ad absurdum). In two and more dimensions, this argument fails, as the function

shows. Its only critical point is at (0,0), which is a local minimum with f(0,0) = 0. However, it cannot be a global one, because f(4,1) = −11