PHAGWARA (PUNJAB)

SESSION 2010-2011

TERM PAPER

ENGINEERING MATHAMETICS-I MTH101

Topic: Comment on maxima and minima of functions, how this concept is helpful in daily life.

DOA: 03/09/10

DOR: 21/09/10

DOS: 15/11/10

Submitted to: Submitted by:

Ms… Mr. …

Deptt. Of Mathametics Roll. No. …

Reg.No…

Class……E6001...

It is not until you under take a project like this one that you realize how massive the effort it really is, or how much you must really upon the self less effort and good will of other . There are many who helped in this project, and I want to thanks them all .It is my pleasure to thank all those who helped me directly or indirectly in presentation of this project .The development of a project of this nature would not have possible without the help of different persons .I am intended to all of them.

I express my deep gratitude to Mrs. Gurpreet (Lecturer of lovely professional university) for helping me and for their continuing support at the every stage of the development of this project by providing sufficient time in the study centre lab.

At the last but not least I am most thankful to all friends and family members for all the encouragement and facilities provided by them which has helped me the most to complete this project work.

Manoj

Introduction

Functions of more than one variable

Maxima and Minima of Functions of Two Variables

The Second Derivative Test for Functions of Two Variables

Maxima and Minima in a Bounded Region

Maxima and Minima for Functions of More than 2 Variables

How is maxima and minima use full in daily life?

References

Definitions of Maxima and minima: In mathematics, maxima and minima, known collectively as extrema (singular: extremum), 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). ...

Functions of more than one variable

For functions of more than one variable, similar conditions apply. 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.In contrast, there are substantial differences between functions of one variable and functions of more than one variable in the identification of global extrema. 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 ƒ(0,0) = 0. However, it cannot be a global one, because ƒ(4,1) = −11.

The global maximum is the point at the top Counterexample

Maxima and minima are more generally defined for sets. In general, if an ordered set S has a greatest element m, m is a maximal element. Furthermore, if S is a subset of an ordered set T and m is the greatest element of S with respect to order induced by T, m is a least upper bound of S in T. The similar result holds for least element, minimal element and greatest lower bound.In the case of a general partial order, the least element (smaller than all other) should not be confused with a minimal element (nothing is smaller). Likewise, a greatest element of a partially ordered set (poset) is an upper bound of the set which is contained within the set, whereas a maximal element m of a poset A is an element of A such that if m ≤ b (for any b in A) then m = b. Any least element or greatest element of a poset is unique, but a poset can have several minimal or maximal

elements. If a poset has more than one maximal element, then these elements will not be mutually comparable.In a totally ordered set, or chain, all elements are mutually comparable, so such a set can have at most one minimal element and at most one maximal element. Then, due to mutual comparability, the minimal element will also be the least element and the maximal element will also be the greatest element. Thus in a totally ordered set we can simply use the terms minimum and maximum. If a chain is finite then it will always have a maximum and a minimum. If a chain is infinite then it need not have a maximum or a minimum. For example, the set of natural numbers has no maximum, though it has a minimum. If an infinite chain S is bounded, then the closure Cl (S) of the set occasionally has a minimum and a maximum, in such case they are called the greatest lower bound and the least upper bound of the set S, respectively.Local Maximum and MinimumFunctions can have "hills and valleys": places where they reach a minimum or maximum value. It may not be the minimum or maximum for the whole function, but locally it is.

Local Maximum First we need to choose an interval:

Then we can say that a local maximum is the point where:The height of the function at "a" is greater than (or equal to) the height anywhere else in that interval.Then we can say that a local maximum is the point where:The height of the function at "a" is greater than (or equal to) the height anywhere else in that interval.Or, more briefly:f(a) ≥ f(x) for all x in the intervalIn other words, there is no height greater than f(a).Note: f(a) should be inside the interval, not at one end or the other.

Local Minimum

Likewise, a local minimum is:f(a) ≤ f(x) for all x in the intervalThe plural of Maximum is MaximaThe plural of Minimum is MinimaMaxima and Minima are collectively called Extrema.

Global (or Absolute) Maximum and MinimumThe maximum or minimum over the entire function is called an "Absolute" or "Global" maximum or minimum.There is only one global maximum (and one global minimum) but there can be more than one local maximum or minimum.

The terms maxima and minima refer to extreme values of a function, that is, the maximum and minimum values that the function attains. Maximum means upper bound or largest possible quantity. The absolute maximum of a function is the largest number contained in the range of the function. That is, if f(a) is greater than or equal to f(x), for all x in the domain of the function, then f(a) is the absolute maximum. For example, the function f(x) = -16x2 + 32x + 6 has a maximum value of 22 occurring at x = 1. Every value of x produces a value of the function that is less than or equal to 22, hence, 22 is an absolute maximum. In terms of its graph, the absolute maximum of a function is the value of the function that corresponds to the highest point on the graph. Conversely, minimum means lower bound or least possible quantity. The absolute minimum of a function is the smallest number in its range and corresponds to the value of the function at the lowest point of its graph. If f(a) is less than or equal to f(x), for all x in the domain of the function, then f(a) is an absolute minimum. As an example, f(x) = 32x2 - 32x - 6 has an absolute minimum of -22, because every value of x produces a value greater than or equal to -22.In some cases, a function will have no absolute maximum or minimum. For instance the function f(x) = 1/x has no absolute maximum value, nor does f(x) = -1/x have an absolute minimum. In still other cases, functions may have relative (or local) maxima and minima. Relative means relative to local or

nearby values of the function. The terms relative maxima and relative minima refer to the largest, or least, value that a function takes on over some small portion or interval of its domain. Thus, if f(b) is greater than or equal to f(b ± h) for small values of h, then f(b) is a local maximum; if f(b) is less than or equal to f(b ± h), then f(b) is a relative minimum. For example, the function f(x) = x4 -12x3 - 58x2 + 180x + 225 has two relative minima (points A and C), one of which is also the absolute minimum (point C) of the function. It also has a relative maximum (point B), but no absolute maximum.Finding the maxima and minima, both absolute and relative, of various functions represents an important class of problems solvable by use of differential calculus. The theory behind finding maximum and minimum values of a function is based on the fact that the derivative of a function is equal to the slope of the tangent. When the values of a function increase as the value of the independent variable increases, the lines that are tangent to the graph of the function have positive slope, and the function is said to be increasing. Conversely, when the values of the function decrease with increasing values of the independent variable, the tangent lines have negative slope, and the function is said to be decreasing. Precisely at the point where the function changes from increasing to decreasing or from decreasing to increasing, the tangent line is horizontal (has slope 0), and the derivative is zero. (With reference to figure 1, the function is decreasing to the left of point A, as well as between points B and C, and increasing between points A and B and to the right of point C). In order to find maximum and minimum points, first find the values of the independent variable for which the derivative of the function is zero, then substitute them in the original function to obtain the corresponding maximum or minimum values of the function. Second, inspect the behavior of the derivative to the left and right of each point. If the derivative Figure 1. Illustration by Hans & Cassidy. Courtesy of Gale Group. is negative on the left and positive on the right, the point is a minimum. If the derivative is positive on the left and negative on the right, the point is a maximum. Equivalently, find

the second derivative at each value of the independent variable that corresponds to a maximum or minimum; if the second derivative is positive, the point is a minimum, if the second derivative is negative the point is a maximum.

Maxima and Minima of Functions of Two Variables

The problem of determining the maximum or minimum of function is encountered in geometry, mechanics, physics, and other fields, and was one of the motivating factors in the development of the calculus in the seventeenth century. Let us recall the procedure for the case of a function of one variable y=f(x). First, we determine points x _c where f'(x)=0. These points are called critical points. At critical points the tangent line is horizontal. This is shown in the figure below. The second derivative test is employed to determine if a critical point is a relative maximum or a relative minimum. If f''(x_ c)>0, then x_ c is a relative minimum. If f''(x_ c)<0, then x_ c is a maximum. If f''(x_ c)=0, then the test gives no information. The notions of critical points and the second derivative test carry over to functions of two variables. Let z=f ( x, y ). Critical points are points in the xy - plane where the tangent plane is horizontal.

Since the normal vector of the tangent plane at (x, y) is given by fx(x, y)i + fy(x, y)j - k

The tangent plane is horizontal if its normal vector points in the z direction. Hence, critical points are solutions of the equations:

fx(x,y) = 0 & fy (x, y) = 0because horizontal planes have normal vector parallel to z-axis. The two equations above must be solved simultaneously.

The Second Derivative Test for Functions of Two Variables

How can we determine if the critical points found above are relative maxima or minima? We apply a second derivative test for functions of two variables. Let (xc, yc) be a critical point and define D(xc,yc) = fxx(xc, yc) fyy(xc , yc) – [fxy (xc, yc)]2

We have the following cases: If D>0 and f_x x(x_ c,y_ c)<0, then f(x, y) has a relative maximum at (x_ c,y _ c). If D>0 and f_ xx(x_ c,y_ c)>0, then f(x,y) has a relative minimum at (x_c,y_c). If D<0, then f(x, y) has a saddle point at (x_ c,y_ c ). zx D=0, the second derivative test is inconclusive.

Maxima and Minima in a Bounded Region

Suppose that our goal is to find the global maximum and minimum of our model function above in the square -2<=x<=2 and -2<=y<=2? There are three types of points that can potentially be global maxima or minima:Relative extrema in the interior of the square. Relative extrema on the boundary of the square. Corner Points. We have already done step 1. There are extrema at (1,0) and (-1,0). The boundary of square consists of 4 parts. Side 1 is y=-2 and -2<=x<=2. On this side, we have z = f(x,-2)= exp{1/3x3 + x –(-2)2 } = g(x) The original function of 2 variables is now a function of x only. We set g'(x)=0 to determine relative extrema on Side 1. It can be shown that x=1 and x=-1 are the relative extrema. Since y=-2, the relative extrema on Side 1 are at (1,-2) and (-1,-2).

On Side 2 (x=-2 and -2<=y<=2) z = f(-2,y)=exp{-1/3(-2)3-2 –y2}=h(y)We set h'(y)=0 to determine the relative extrema. It can be shown that y=0 is the only critical point, corresponding to (-2,0). We play the same game to determine the relative extrema on the other 2 sides. It can be shown that they are (2,0), (1,2), and (-1,2). Finally, we must include the 4 corners (-2,-2), (-2,2), (2,-2), and (2,2). In summary, the candidates for global maximum and minimum are (-1,0), (1,0), (1,-2), (-1,-2), (-2,0), (2,0), (1,2), (-1,2), (-2,-2), (-2,2), (2,-2), and (2,2). We evaluate f(x,y) at each of these points to determine the global max and min in the square. The global maximum occurs (-2,0) and (1,0). This can be seen in the figure above. The global minimum occurs at 4 points: (-1,2), (-1,-2), (2,2), and (2,-2).

Example: Maxima and Minima in a Disk

Another example of a bounded region is the disk of radius 2 centered at the origin. We proceed as in the previous example, determining in the 3 classes above. (1,0) and (-1,0) lie in the interior of the disk. The boundary of the disk is the circle x^2+y^2=4. To find extreme points on the disk we parameterize the circle. A natural parameterization is x=2cos(t) and y=2sin(t) for 0<=t<=2*pi. We substitute these expressions into z=f(x,y) and obtain z = f(x, y) = f( cos(t),sin(t)) = exp (-8/3 cos3t + 2 cos t – 4sin2t) = g(t)On the circle, the original functions of 2 variables is reduced to a function of 1 variable. We can determine the extrema on the circle using techniques from calculus of on variable. In this problem there are not any corners. Hence, we determine the global max and min by considering points in the interior of the disk and on the circle. An alternative method for finding the maximum and minimum on the circle is the method of Lagrange multipliers.

Maxima and Minima for Functions of More than 2 Variables

The notion of extreme points can be extended to functions of more than 2 variables. Suppose z=f(x_1,x_2,...,x_ n). (a_1,a_2,...,a_ n) is extreme point if it satisfies the n equations

(x1,x2,……,xn) = 0 i = 1,2,3,……..,nThere is not a general second derivative test to determine if a point is a relative maximum or minimum for functions of more than two variables.

How is maxima and minima use full in daily life?

Maxima and minima pop up all over the place in our daily lives. They can be found anywhere we are interested in the highest and/or lowest value of a given system; if you look hard enough, you can probably find them just about anywhere! Here are just a few examples of where you might encounter maxima and minima:

A meteorologist creates a model that predicts temperature variance with respect to time. The absolute maximum and minimum of this function over any 24-hour period are the forecasted high and low temperatures, as later reported on The Weather Channel or the evening news.

The director of a theme park works with a model of total revenue as a function of admission price. The location of the absolute maximum of this function represents the ideal admission price. An actuary works with functions that represent the probability of various negative events occurring. The local minima of these functions correspond to lucrative markets for his/her insurance company – low-risk, high-reward ventures.

Higher Engineering Mathematics by B V Ramana

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