Maxima and Minima of Functions

Maxima and Minima of FunctionsThis term opus presents concise explanations of the subjects general principles and de qualitys worked examples freely to expand the ideas slightly solving the problems by sui delay methods. Each example shows the method of obtaining the resolve and includes additional explanatory techniques. For some topics, where it would have been difficult to understand a solution given on a single problem, the solution has been drawn in step-by-step form. All the figures habitd have been taken from Google Book search.The term paper covers the necessary definitions on MAXIMA AND MINIMA OF THE FUNCTIONS and some of its important applications. It covers the topic such as types of other method for solving the big problem in a crosscut method known . The aspects of how to develop some of the most comm just now seen problems is also covered in this term paper. The motive of this term paper is make the indorser familiar with the concepts of application of maxima and minima of the voice andwhere this is used. Focus has been more on taking the simpler problem so that the concept could be made cle arr even to the beginners to engineering mathematics.MAXIMA AND MINIMAThe draw below shows part of a involvement y = f(x).The bakshis A is a local anesthetic maximal and the allude B is a local marginal.At each of these points the tangent to the curve is parallel to theX axis so the derivative of the function is zero. Both of these pointsare therefore stationary points of the function. The term local is usedsince these points are the supreme and negligible in this particularRegion.The rate of change of a function is measured by its derivative.When the derivative is positive, the function is increasing,When the derivative is negative, the function is decreasing.Thus the rate of change of the gradient is measured by its derivative,Which is the second derivative of the original function?Functions can have hills and valleys places where they rea ch a marginal or maximum quantify.It may not be the minimum or maximum for the whole function, moreover locally it is.You can see where they are,but how do we define them? topical anesthetic Maximum first off we need to choose an detachmentThen we can say that a local maximum is the point whereThe height of the function at a is greater than (or equal to) the height anywhere else in that legal separation.Or, more brieflyf(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 MinimumLikewise, a local minimum isf(a) f(x) for all x in the intervalThe plural of Maximum is MaximaThe plural of Minimum is MinimaMaxima and Minima are collectively called Extremaworld(a) (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 th an one local maximum or minimum. assuming this function continues downwards to left and rightThe Global Maximum is about 3.7The Global Minimum is -InfinityMaxima and Minima of Functions of Two VariablesLocate coition maxima, minima and saddle points of functions of two variables. some(prenominal) examples with elaborated solutions are presented. 3-Dimensional graphs of functions are shown to confirm the existence of these points. More on Optimization Problems with Functions of Two Variables in this web site.TheoremLet f be a function with two variables with continuous second order partial derivativesfxx, fyyand fxyat a life-sustaining point (a,b). LetD = fxx(a,b) fyy(a,b) fxy2(a,b)If D 0 and fxx(a,b) 0, then f has a sexual congress minimum at (a,b).If D 0 and fxx(a,b) If D If D = 0, then no conclusion can be drawn.We now present several examples with detailed solutions on how to locate relative minima, maxima and saddle points of functions of two variables. When too many criti cal points are found, the use of a table is very convenient. guinea pig 1 sterilise the critical points and locate any relative minima, maxima and saddle points of function f specify byf(x , y) = 22+ 2xy + 2y2- 6xSolution to Example 1 nonplus the prototypic partial derivatives fxand fy.fx(x,y) = 4x + 2y 6fy(x,y) = 2x + 4yThe critical points satisfy the equations fx(x,y) = 0 and fy(x,y) = 0 simultaneously. Hence.4x + 2y 6 = 02x + 4y = 0The above system of equations has one solution at the point (2,-1).We now need to see the second order partial derivatives fxx(x,y), fyy(x,y) and fxy(x,y).fxx(x,y) = 4fxx(x,y) = 4fxy(x,y) = 2We now need to find D defined above.D = fxx(2,-1) fyy(2,-1) fxy2(2,-1) = ( 4 )( 4 ) 22= 12Since D is positive and fxx(2,-1) is also positive, according to the above theorem function f has a local minimum at (2,-1).The 3-Dimensional graph of function f given above shows that f has a local minimum at the point (2,-1,f(2,-1)) = (2,-1,-6).Example 2Determine the critical points and locate any relative minima, maxima and saddle points of function f defined byf(x , y) = 22- 4xy + y4+ 2Solution to Example 2Find the first partial derivatives fxand fy.fx(x,y) = 4x 4yfy(x,y) = 4x + 4y3Determine the critical points by solving the equations fx(x,y) = 0 and fy(x,y) = 0 simultaneously. Hence.4x 4y = 0 4x + 4y3= 0The first equation gives x = y. Substitute x by y in the equation 4x + 4y3= 0 to obtain. 4y + 4y3= 0Factor and solve for y.4y(-1 + y2) = 0y = 0 , y = 1 and y = -1We now use the equation x = y to find the critical points.(0 , 0) , (1 , 1) and (-1 , -1)We now determine the second order partial derivatives.fxx(x,y) = 4fyy(x,y) = 12y2fxy(x,y) = -4We now use a table to study the signs of D and fxx(a,b) and use the above theorem to decide on whether a given critical point is a saddle point, relative maximum or minimum.critical point (a,b)(0,0)(1,1)(-1,1)fxx(a,b)444fyy(a,b)01212fxy(a,b)-4-4-4D-163232saddle pointrelative minimumrelative minimumA 3-Dimensional graph of function f shows that f has two local minima at (-1,-1,1) and (1,1,1) and one saddle point at (0,0,2).Example 3Determine the critical points and locate any relative minima, maxima and saddle points of function f defined byf(x , y) = x4- y4+ 4xy.Solution to Example 3First partial derivatives fxand fyare given by.fx(x,y) = 43+ 4yfy(x,y) = 4y3+ 4xWe now solve the equations fy(x,y) = 0 and fx(x,y) = 0 to find the critical points.. 43+ 4y = 0 4y3+ 4x = 0The first equation gives y = x3. Combined with the second equation, we obtain. 4(x3)3+ 4x = 0Which may be written as .x(x4- 1)(x4+ 1) = 0Which has the solutions.x = 0 , -1 and 1.We now use the equation y = x3to find the critical points.(0 , 0) , (1 , 1) and (-1 , -1)We now determine the second order partial derivatives.fxx(x,y) = -122The First Derivative Maxima and MinimaConsider the function f(x)=3443122+3 on the interval 23. We cannot find regions of which f is increasing or decreasing, relative maxima or minim a, or the compulsive maximum or minimum appreciate of f on 23 by inspection. Graphing by hand is tedious and imprecise. Even the use of a graphing program will only give us an approximation for the locations and values of maxima and minima. We can use the first derivative of f, however, to find all these things quickly and easily.Increasing or Decreasing?Let f be continuous on an interval I and differentiable on the interior of I.If f(x)0 for all xI, then f is increasing on I.If f(x)0 for all xI, then f is decreasing on I.ExampleThe function f(x)=3443122+3 has first derivative f(x)===12312224x12x(x2x2)12x(x+1)(x2) Thus, f(x) is increasing on (10)(2) and decreasing on (1)(02).Relative Maxima and MinimaRelative extrema of f occur at critical points of f, values x0 for which either f(x0)=0 or f(x0) is undefined.First Derivative TestSuppose f is continuous at a critical point x0.If f(x)0 on an open interval extending left from x0 and f(x)0 on an open interval extending right from x0, then f has a relative maximum at x0.If f(x)0 on an open interval extending left from x0 and f(x)0 on an open interval extending right from x0, then f has a relative minimum at x0.If f(x) has the same sign on both an open interval extending left from x0 and an open interval extending right from x0, then f does not have a relative vizor at x0.In summary, relative extrema occur where f(x) changes sign.ExampleOur function f(x)=3443122+3 is differentiable everywhere on 23, with f(x)=0 for x=102. These are the three critical points of f on 23. By the First Derivative Test, f has a relative maximum at x=0 and relative minima at x=1 and x=2.Absolute Maxima and MinimaIf f has an extreme value on an open interval, then the extreme value occurs at a critical point of f.If f has an extreme value on a closed interval, then the extreme value occurs either at a critical point or at an endpoint.According to the Extreme Value Theorem, if a function is continuous on a closed interval, then it achiev es both an absolute maximum and an absolute minimum on the interval.ExampleSince f(x)=3443122+3 is continuous on 23, f must have an absolute maximum and an absolute minimum on 23. We simply need to check the value of f at the critical points x=102 and at the endpoints x=2 and x=3 f(2)f(1)f(0)f(2)f(3)=====35232930 Thus, on 23, f(x) achieves a maximum value of 35 at x=2 and a minimum value of -29 at x=2.We have discovered a lot about the shape of f(x)=3443122+3 without ever graphing it at one time take a look at the graph and verify each of our conclusions.APPLICATION AND CONCLUSIONThe 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 maxi mum. For example, the function f(x) = -162 + 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) = 322 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 authority the function f(x) = 1/x has no absolute maximum value, nor does f(x) = -1/x have an absolut e 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 -123 582 + 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 maxima or minima also has important applications in analogue algebra and game theory. For example, linear programming consists of maximizing (or minimizing) a particular quantity while requiring that certain constraints be imposed on other quantities. The q uantity to be maximized (or minimized), as intumesce as each of the constraints, is represented by an equation or inequality. The resulting system of equations or inequalities, usually linear, often contains hundreds or thousands of variables. The idea is to find the maximum value of a particular variable that represents a solution to the whole system. A practical example might be minimizing the cost of producing an automobile given certain known constraints on the cost of each part, and the time spent by each laborer, all of which may be interdependent. Regardless of the application, though, the key step in any maxima or minima problem is expressing the problem in mathematical terms.