Calculus
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The document discusses methods for finding the maximum and minimum values of functions with two independent variables, including: 1) Using the second derivative test to determine if a critical point is a local maximum or minimum. 2) Finding absolute maxima and minima, which are the overall largest and smallest function values in the domain. 3) Identifying stationary points where the first partial derivatives are equal to zero. 4) Using the Hessian matrix and eigenvalues to generalize the second derivative test to functions of more than one variable.
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1.1 Maxima & Minima . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.1 Maximum & Minimum Function . . . . . . . . . . . . . 3
1.2 Some Definitions . . . . . . . . . . . . . . . . . . . . . . . . . 4
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reasons for changing variables but the main reason for changing variables is to convert the
integrand into something simpler and also to transform the region into another region which is
easy to work with. When we convert into a new set of variables it is not always easy to find the
limits. So, before we move into changing variables with multiple integrals we first need to see
how the region may change with a change of variables. In order to change variables in an
integration we will need the Jacobian of the transformation.
Group No 05, calculus.pptx
1) The document introduces concepts of differential calculus including derivatives, limits, continuity, and fundamental rules of taking derivatives.
2) It provides examples of calculating derivatives using notations like delta f, limits, and various derivative rules including the power rule, product rule, and quotient rule.
3) Methods for finding local maxima and minima are discussed, including using the first derivative test and analyzing stationary points where the first derivative is zero based on whether the derivative is increasing or decreasing on both sides.
WEEK-4-Piecewise-Function-and-Rational-Function.pptx
This document discusses various topics related to piecewise functions and rational functions:
- It defines piecewise functions and provides examples of evaluating piecewise functions at given values.
- It introduces rational functions as functions of the form p(x)/q(x) where p(x) and q(x) are polynomials and q(x) is not equal to zero. It discusses representing rational functions in different forms.
- It explains how to identify restrictions or extraneous roots of rational functions by setting the denominator equal to zero. It also discusses how to determine the domain of a rational function based on its restrictions.
- Finally, it defines vertical and horizontal asymptotes of rational functions. It provides
Module 3 polynomial functions
This module discusses polynomial functions of degree greater than two. The key points are:
1. The graph of a third-degree polynomial has both a minimum and maximum point, while higher degree polynomials have one less turning point than their degree.
2. Methods like finding upper and lower bounds and Descartes' Rule of Signs can help determine properties of the graph like zeros.
3. Odd degree polynomials increase on the far left and right if the leading term is positive, and decrease if negative. Even degree polynomials increase on the far left and decrease on the far right, or vice versa.
maxima & Minima thoeyr&solved.Module-4pdf
In this chapter we shall study those points of the domain of a function where its graph changes its direction from upwards to downwards or from downwards to upwards. At such points the derivative of the function, if it exists, is necessarily zero.
The value of a function f (x) is said to be maximum at x = a, if there exists a very small positive number h, such that
f(x) < f(a) x (a – h,a + h) , x a
In this case the point x = a is called a point of maxima for the function f(x).
Simlarly, the value of f(x) is said to the minimum at x = b, If there exists a very small positive number, h, such that
f(x) > f(b), x (b – h,b + h), x b
In this case x = b is called the point of minima for the function f(x).
Hene we find that,
(i) x = a is a maximum point of f(x)
f(a) f(a h) 0
f(a) f(a h) 0
(ii) x = b is a minimum point of f(x)
Rf(b) f(b h) 0
f(b) f(b h) 0
(iii) x = c is neither a maximum point nor a minimum point
Note :
(i) The maximum and minimum points are also known as extreme points.
(ii) A function may have more than one maximum and minimum points.
(iii) A maximum value of a function f(x) in an interval [a,b] is not necessarily its greatest value in that interval. Similarly, a minimum value may not be the least value of the function. A minimum value may be greater than some maximum value for a function.
(iv) If a continuous function has only one maximum (minimum) point, then at this point function has its greatest (least) value.
(v) Monotonic functions do not have extreme points.
Ex. Function y = sin x, x (0, ) has a maximum point at x = /2 because the value of sin /2 is greatest in the given interval for sin x.
Clearly function y = sin x is increasing in the interval (0, /2) and decreasing in the interval ( /2, ) for that reason also it has maxima at x = /2. Similarly we can see from the graph of cos x which has a minimum point at x = .
Ex. f(x) = x2 , x (–1,1) has a minimum point at x = 0 because at x = 0, the value of x2 is 0, which is
less than the all the values of function at different points of the interval.
Clearly function y = x2 is decreasing in the interval
Rf(c) f(c h)
Sand
Tf(c) f(c h) 0
|V| have opposite signs.
(–1, 0) and increasing in the interval (0,1) So it has minima at x = 0.
Ex. f(x) = |x| has a minimum point at x = 0. It can be easily observed from its graph.
A. Necessary Condition : A point x = a is an extreme point of a function f(x) if f’(a) = 0, provided f’(a) exists. Thus if f’ (a) exists, then
x = a is an extreme point f’(a) = 0
or
f’ (a) 0 x = a is not an extreme point.
But its converse is not true i.e.
f’ (a) = 0 x = a is an extreme point.
For example if f(x) = x3 , then f’ (0) = 0 but x = 0 is not an extreme point.
B. Sufficient Condition :
(i) The value of the function f(x) at x = a is maximum, if f’ (a) = 0 and f” (a) < 0.
(ii) The value of
Curve sketching
This document outlines the 7 steps for sketching the curve of a function: 1) Determine the domain, 2) Find critical points, 3) Determine graph direction and max/min, 4) Use the second derivative to find concavity and points of inflection, 5) Find asymptotes, 6) Find intercepts and important points, 7) Combine evidence to graph the function. Key tests are outlined for max/min, concavity, and points of inflection using the first and second derivatives.
Graphing polynomials
This document discusses how to graph polynomial functions and find local extrema. It provides instructions on making a table of values, plotting points, connecting them with a smooth curve, and checking end behavior based on the degree and leading coefficient. Extrema are defined as local maxima or minima, where the graph changes from increasing to decreasing. Examples are given to demonstrate graphing polynomials and finding turning points that indicate local extrema.
Mac2311 study guide-tcm6-49721
This document provides an overview of key calculus concepts and formulas taught in a Calculus I course at Miami Dade College - Hialeah Campus. The topics covered include limits and derivatives, integration, optimization techniques, and applications of calculus to economics, business, physics, and other fields. The document is intended as a study guide for students in the Calculus I class taught by Professor Mohammad Shakil.
C2 differentiation jan 22
The document discusses increasing and decreasing functions. An increasing function has a positive gradient, while a decreasing function has a negative gradient. Some functions can be increasing over one interval and decreasing over another. You need to be able to determine the intervals where a function is increasing or decreasing by examining its gradient. The document provides examples of finding where a function is decreasing by taking the derivative, setting it equal to 0, and solving for the range of x-values that make the gradient negative. It also discusses using derivatives to find the coordinates of stationary points like maxima and minima, and using the second derivative to determine if a stationary point is a maximum or minimum.
Ap calculus extrema v2
The document discusses extreme values (maximums and minimums) of functions. It defines local and absolute extrema and notes that not all functions have extrema. Critical points, where the derivative is zero or undefined, are discussed. The relationship between critical points and extrema is explained by Fermat's Theorem. Methods for finding extrema on closed intervals using critical points and endpoints are presented, along with an example. Rolle's Theorem relating critical points to functions with equal values at endpoints is also introduced.
Profit Maximization (stats).ppt
This document outlines the steps to find the maximum value of a function:
1. Take the derivative of the function to find critical points where it is increasing, decreasing, or constant.
2. Check the critical points by taking the second derivative to determine if they are local maximums or minimums.
3. Plug the critical points that are maximums back into the original function to find the maximum value.
Calc 3.4
The document discusses using the second derivative test to determine the concavity and points of inflection of functions. It explains that the second derivative test involves finding intervals where the second derivative is positive or negative to determine if a graph is concave up or down. Points of inflection occur when the second derivative is equal to zero. Examples are provided to demonstrate how to apply this test to locate intervals of concavity and points of inflection of various functions.
5.2 first and second derivative test
This document discusses methods for finding relative extrema of functions:
1. The First Derivative Test (FDT) states that a critical point is a relative maximum if the derivative changes from positive to negative, and a relative minimum if the derivative changes from negative to positive.
2. The Second Derivative Test (SDT) states that a critical point is a relative maximum if the second derivative is negative, and a relative minimum if the second derivative is positive.
3. Examples are provided to demonstrate using the FDT and SDT to find relative extrema of functions.
2 random variables notes 2p3
The document defines and discusses random variables. It begins by defining a random variable as a function that assigns a real number to each outcome of a random experiment. It then discusses the conditions for a function to be considered a random variable. The document outlines the key types of random variables as discrete, continuous, and mixed and introduces the cumulative distribution function (CDF) and probability density function (PDF) as ways to describe the distribution of a random variable. It provides examples of CDFs and PDFs for discrete random variables and discusses properties of distribution and density functions. The document also introduces important continuous random variables like the Gaussian random variable.
0.5.derivatives
The document outlines key calculus concepts including:
- Functions, derivatives, differentiation rules, and the definition of a derivative as an infinitesimal change in a function with respect to a variable.
- Concepts related to derivatives such as local/absolute extrema, critical points, increasing/decreasing functions, concavity, asymptotes, and inflection points.
- How to use the first and second derivative tests to determine local extrema, concavity, and increasing/decreasing behavior.
3.1 Functions and Function Notation
* Determine whether a relation represents a function.
* Find the value of a function.
* Determine whether a function is one-to-one.
* Use the vertical line test to identify functions.
* Graph the functions listed in the library of functions.
3.1 Characteristics of Polynomial Functions.pptx
This document provides information about polynomial functions including:
- Their general form as a sum of terms with variables raised to whole number powers and real number coefficients.
- Characteristics like degree, leading coefficient, constant term, and their relationship to the graph.
- Examples of graphs of polynomial functions of varying degrees and their characteristics like possible intercepts and behavior at endpoints.
- Worked examples identifying characteristics of given polynomial functions.
Index cards characteristics and theorems
The document defines and explains concepts related to concavity, points of inflection, absolute extrema, increasing and decreasing intervals, local extrema, and theorems such as the Mean Value Theorem, Rolle's Theorem, the Second Derivative Test, the Extreme Value Theorem, the Intermediate Value Theorem, and Fermat's Theorem. It provides definitions, steps to find or determine each concept, and how the theorems relate to finding critical points and values of functions.
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留信网认证的作用:
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2:同时对留学生所学专业登记给予评定
3:国家专业人才认证中心颁发入库证书
4:这个认证书并且可以归档倒地方
5:凡事获得留信网入网的信息将会逐步更新到个人身份内，将在公安局网内查询个人身份证信息后，同步读取人才网入库信息
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8:在国家人才网主办的国家网络招聘大会中纳入资料，供国家高端企业选择人才
→ 【关于价格问题（保证一手价格）
我们所定的价格是非常合理的，而且我们现在做得单子大多数都是代理和回头客户介绍的所以一般现在有新的单子 我给客户的都是第一手的代理价格，因为我想坦诚对待大家 不想跟大家在价格方面浪费时间
对于老客户或者被老客户介绍过来的朋友，我们都会适当给一些优惠。
选择实体注册公司办理，更放心，更安全！我们的承诺：可来公司面谈，可签订合同，会陪同客户一起到教育部认证窗口递交认证材料，客户在教育部官方认证查询网站查询到认证通过结果后付款，不成功不收费！
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校徽：象征着学校的荣誉和传承。
校名:学校英文全称
授予学位：本部分将注明获得的具体学位名称。
毕业生姓名：这是最重要的信息之一，标志着该证书是由特定人员获得的。
颁发日期：这是毕业正式生效的时间，也代表着毕业生学业的结束。
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◇申请留学生创业基金
留信网认证的作用:
1:该专业认证可证明留学生真实身份
2:同时对留学生所学专业登记给予评定
3:国家专业人才认证中心颁发入库证书
4:这个认证书并且可以归档倒地方
5:凡事获得留信网入网的信息将会逐步更新到个人身份内，将在公安局网内查询个人身份证信息后，同步读取人才网入库信息
6:个人职称评审加20分
7:个人信誉贷款加10分
8:在国家人才网主办的国家网络招聘大会中纳入资料，供国家高端企业选择人才
办理(uofo毕业证书)美国俄勒冈大学毕业证【微信：741003700 】外观非常简单，由纸质材料制成，上面印有校徽、校名、毕业生姓名、专业等信息。
办理(uofo毕业证书)美国俄勒冈大学毕业证【微信：741003700 】格式相对统一，各专业都有相应的模板。通常包括以下部分：
校徽：象征着学校的荣誉和传承。
校名:学校英文全称
授予学位：本部分将注明获得的具体学位名称。
毕业生姓名：这是最重要的信息之一，标志着该证书是由特定人员获得的。
颁发日期：这是毕业正式生效的时间，也代表着毕业生学业的结束。
其他信息：根据不同的专业和学位，可能会有一些特定的信息或章节。
办理(uofo毕业证书)美国俄勒冈大学毕业证【微信：741003700 】价值很高，需要妥善保管。一般来说，应放置在安全、干燥、防潮的地方，避免长时间暴露在阳光下。如需使用，最好使用复印件而不是原件，以免丢失。
综上所述，办理(uofo毕业证书)美国俄勒冈大学毕业证【微信：741003700 】是证明身份和学历的高价值文件。外观简单庄重，格式统一，包括重要的个人信息和发布日期。对持有人来说，妥善保管是非常重要的。
Tools & Techniques for Commissioning and Maintaining PV Systems W-Animations ...
Join us for this solutions-based webinar on the tools and techniques for commissioning and maintaining PV Systems. In this session, we'll review the process of building and maintaining a solar array, starting with installation and commissioning, then reviewing operations and maintenance of the system. This course will review insulation resistance testing, I-V curve testing, earth-bond continuity, ground resistance testing, performance tests, visual inspections, ground and arc fault testing procedures, and power quality analysis.
Fluke Solar Application Specialist Will White is presenting on this engaging topic:
Will has worked in the renewable energy industry since 2005, first as an installer for a small east coast solar integrator before adding sales, design, and project management to his skillset. In 2022, Will joined Fluke as a solar application specialist, where he supports their renewable energy testing equipment like IV-curve tracers, electrical meters, and thermal imaging cameras. Experienced in wind power, solar thermal, energy storage, and all scales of PV, Will has primarily focused on residential and small commercial systems. He is passionate about implementing high-quality, code-compliant installation techniques.
一比一原版(osu毕业证书)美国俄勒冈州立大学毕业证如何办理
原版一模一样【微信：741003700 】【(osu毕业证书)美国俄勒冈州立大学毕业证成绩单】【微信：741003700 】学位证，留信认证（真实可查，永久存档）原件一模一样纸张工艺/offer、雅思、外壳等材料/诚信可靠,可直接看成品样本，帮您解决无法毕业带来的各种难题！外壳，原版制作，诚信可靠，可直接看成品样本。行业标杆！精益求精，诚心合作，真诚制作！多年品质 ,按需精细制作，24小时接单,全套进口原装设备。十五年致力于帮助留学生解决难题，包您满意。
本公司拥有海外各大学样板无数，能完美还原。
1:1完美还原海外各大学毕业材料上的工艺：水印，阴影底纹，钢印LOGO烫金烫银，LOGO烫金烫银复合重叠。文字图案浮雕、激光镭射、紫外荧光、温感、复印防伪等防伪工艺。材料咨询办理、认证咨询办理请加学历顾问Q/微741003700
【主营项目】
一.毕业证【q微741003700】成绩单、使馆认证、教育部认证、雅思托福成绩单、学生卡等！
二.真实使馆公证(即留学回国人员证明,不成功不收费)
三.真实教育部学历学位认证（教育部存档！教育部留服网站永久可查）
四.办理各国各大学文凭(一对一专业服务,可全程监控跟踪进度)
如果您处于以下几种情况：
◇在校期间，因各种原因未能顺利毕业……拿不到官方毕业证【q/微741003700】
◇面对父母的压力，希望尽快拿到；
◇不清楚认证流程以及材料该如何准备；
◇回国时间很长，忘记办理；
◇回国马上就要找工作，办给用人单位看；
◇企事业单位必须要求办理的
◇需要报考公务员、购买免税车、落转户口
◇申请留学生创业基金
留信网认证的作用:
1:该专业认证可证明留学生真实身份
2:同时对留学生所学专业登记给予评定
3:国家专业人才认证中心颁发入库证书
4:这个认证书并且可以归档倒地方
5:凡事获得留信网入网的信息将会逐步更新到个人身份内，将在公安局网内查询个人身份证信息后，同步读取人才网入库信息
6:个人职称评审加20分
7:个人信誉贷款加10分
8:在国家人才网主办的国家网络招聘大会中纳入资料，供国家高端企业选择人才
办理(osu毕业证书)美国俄勒冈州立大学毕业证【微信：741003700 】外观非常简单，由纸质材料制成，上面印有校徽、校名、毕业生姓名、专业等信息。
办理(osu毕业证书)美国俄勒冈州立大学毕业证【微信：741003700 】格式相对统一，各专业都有相应的模板。通常包括以下部分：
校徽：象征着学校的荣誉和传承。
校名:学校英文全称
授予学位：本部分将注明获得的具体学位名称。
毕业生姓名：这是最重要的信息之一，标志着该证书是由特定人员获得的。
颁发日期：这是毕业正式生效的时间，也代表着毕业生学业的结束。
其他信息：根据不同的专业和学位，可能会有一些特定的信息或章节。
办理(osu毕业证书)美国俄勒冈州立大学毕业证【微信：741003700 】价值很高，需要妥善保管。一般来说，应放置在安全、干燥、防潮的地方，避免长时间暴露在阳光下。如需使用，最好使用复印件而不是原件，以免丢失。
综上所述，办理(osu毕业证书)美国俄勒冈州立大学毕业证【微信：741003700 】是证明身份和学历的高价值文件。外观简单庄重，格式统一，包括重要的个人信息和发布日期。对持有人来说，妥善保管是非常重要的。
一比一原版(爱大毕业证书)爱荷华大学毕业证如何办理
原版一模一样【微信：741003700 】【(爱大毕业证书)爱荷华大学毕业证成绩单】【微信：741003700 】学位证，留信认证（真实可查，永久存档）原件一模一样纸张工艺/offer、雅思、外壳等材料/诚信可靠,可直接看成品样本，帮您解决无法毕业带来的各种难题！外壳，原版制作，诚信可靠，可直接看成品样本。行业标杆！精益求精，诚心合作，真诚制作！多年品质 ,按需精细制作，24小时接单,全套进口原装设备。十五年致力于帮助留学生解决难题，包您满意。
本公司拥有海外各大学样板无数，能完美还原。
1:1完美还原海外各大学毕业材料上的工艺：水印，阴影底纹，钢印LOGO烫金烫银，LOGO烫金烫银复合重叠。文字图案浮雕、激光镭射、紫外荧光、温感、复印防伪等防伪工艺。材料咨询办理、认证咨询办理请加学历顾问Q/微741003700
【主营项目】
一.毕业证【q微741003700】成绩单、使馆认证、教育部认证、雅思托福成绩单、学生卡等！
二.真实使馆公证(即留学回国人员证明,不成功不收费)
三.真实教育部学历学位认证（教育部存档！教育部留服网站永久可查）
四.办理各国各大学文凭(一对一专业服务,可全程监控跟踪进度)
如果您处于以下几种情况：
◇在校期间，因各种原因未能顺利毕业……拿不到官方毕业证【q/微741003700】
◇面对父母的压力，希望尽快拿到；
◇不清楚认证流程以及材料该如何准备；
◇回国时间很长，忘记办理；
◇回国马上就要找工作，办给用人单位看；
◇企事业单位必须要求办理的
◇需要报考公务员、购买免税车、落转户口
◇申请留学生创业基金
留信网认证的作用:
1:该专业认证可证明留学生真实身份
2:同时对留学生所学专业登记给予评定
3:国家专业人才认证中心颁发入库证书
4:这个认证书并且可以归档倒地方
5:凡事获得留信网入网的信息将会逐步更新到个人身份内，将在公安局网内查询个人身份证信息后，同步读取人才网入库信息
6:个人职称评审加20分
7:个人信誉贷款加10分
8:在国家人才网主办的国家网络招聘大会中纳入资料，供国家高端企业选择人才
办理(爱大毕业证书)爱荷华大学毕业证【微信：741003700 】外观非常简单，由纸质材料制成，上面印有校徽、校名、毕业生姓名、专业等信息。
办理(爱大毕业证书)爱荷华大学毕业证【微信：741003700 】格式相对统一，各专业都有相应的模板。通常包括以下部分：
校徽：象征着学校的荣誉和传承。
校名:学校英文全称
授予学位：本部分将注明获得的具体学位名称。
毕业生姓名：这是最重要的信息之一，标志着该证书是由特定人员获得的。
颁发日期：这是毕业正式生效的时间，也代表着毕业生学业的结束。
其他信息：根据不同的专业和学位，可能会有一些特定的信息或章节。
办理(爱大毕业证书)爱荷华大学毕业证【微信：741003700 】价值很高，需要妥善保管。一般来说，应放置在安全、干燥、防潮的地方，避免长时间暴露在阳光下。如需使用，最好使用复印件而不是原件，以免丢失。
综上所述，办理(爱大毕业证书)爱荷华大学毕业证【微信：741003700 】是证明身份和学历的高价值文件。外观简单庄重，格式统一，包括重要的个人信息和发布日期。对持有人来说，妥善保管是非常重要的。
DEEP LEARNING FOR SMART GRID INTRUSION DETECTION: A HYBRID CNN-LSTM-BASED MODEL
As digital technology becomes more deeply embedded in power systems, protecting the communication
networks of Smart Grids (SG) has emerged as a critical concern. Distributed Network Protocol 3 (DNP3)
represents a multi-tiered application layer protocol extensively utilized in Supervisory Control and Data
Acquisition (SCADA)-based smart grids to facilitate real-time data gathering and control functionalities.
Robust Intrusion Detection Systems (IDS) are necessary for early threat detection and mitigation because
of the interconnection of these networks, which makes them vulnerable to a variety of cyberattacks. To
solve this issue, this paper develops a hybrid Deep Learning (DL) model specifically designed for intrusion
detection in smart grids. The proposed approach is a combination of the Convolutional Neural Network
(CNN) and the Long-Short-Term Memory algorithms (LSTM). We employed a recent intrusion detection
dataset (DNP3), which focuses on unauthorized commands and Denial of Service (DoS) cyberattacks, to
train and test our model. The results of our experiments show that our CNN-LSTM method is much better
at finding smart grid intrusions than other deep learning algorithms used for classification. In addition,
our proposed approach improves accuracy, precision, recall, and F1 score, achieving a high detection
accuracy rate of 99.50%.
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Calculus
- 1. Maxima and Minima Values by Second Derivation Method. • Maxima and Minima of Two Functions of Two Indipendent Variables. • In calculus, the second derivative test is a criterion for determining whether a given critical point of a real function of one variable is a local maxmimum or a local minimum using the value of the second derivative at the point. The test states: if the function f is twice differentiable at a critical point x (i.e. f'(x) = 0), then: If then f has a local maximum at 𝑥 If then f has a local minimum at 𝑥 If then the test is inconclusive.
- 2. • Absolute Mzxima and Minima. The Absolute Maxima and Minima Values of z = f (x,y) indicate the over all largest and smallr=est values of f in function’s domain. • Stationary Points. The points (x,y) for whitch df/dx = 0 and df/dy = 0 are called Stationary Points of f (x , y). • All Points of Local Extrime anr Stationary Points. • But, the Converse may or maynot be true. • Critical Points. Critical Points of f ( x, y) are Those Points Which are Either a Stationary Points or Where or both of fx or fy do not Exist. • Like in the Case of Stationary Points , the Points Where one or both of fx or fy do not Exist , may Have an Extreme Valyue. • Saddle Point or Minimax Point. • In mathematics, a saddle point is a point in the range of a function that is a stationary point but not a local extremum. The name derives from the fact that the prototypical example in two dimensions is a surface that curves up in one direction, and curves down in a different direction, resembling a saddle or a mountain pass. In terms of contour lines, a saddle point in two dimensions gives rise to a contour that appears to intersect itself.
- 3. • Working Rules for Finding Extemum Values of a Function f ( x, y). (a). Find df/dx and df/dy. (b). Find Points ( x, y) for Whitchdf/dx = 0 and df/dy = 0. This Gives the Statiopnary Points. (c). r = 𝑑2𝑓 𝑑𝑥2 , 𝑠 = 𝑑2𝑓 𝑑𝑥𝑑𝑦 , 𝑡 = 𝑑2𝑓 𝑑𝑦2 The Nature of Stationary Points are rt-𝑠2.
- 4. • Multivariable Case. • For a function of more than one variable, the second derivative test generalizes to a test based on the eigenvalues of the function's Hessian matrix at the critical point. In particular, assuming that all second order partial derivatives of f are continuous on a neighbourhood of a critical point x, then if the eigenvalues of the Hessian at x are all positive, then x is a local minimum. If the eigenvalues are all negative, then x is a local maximum, and if some are positive and some negative, then the point is a saddle point. If the Hessian matrix is singular, then the second derivative test is inconclusive. • Concavity Test. • A related but distinct use of second derivatives is to determine whether a function is concave up or concave down at a point. It does not, however, provide information aboutinflection points. Specifically, a twice- differentiable function f is concave up if and concave down if Note that if ,then has zero second derivative, yet is not an inflection point, so the second derivative alone does not give enough information to determine if a given point is an inflection point.
- 5. Examples • Ex.1 Discribe The Nature of The Stationary Points For f(x,y)=𝒙 𝟒 + 𝒚 𝟒 − 𝟐𝒙 𝟐 + 𝟒𝒙𝒚 − 𝟐𝒚 𝟐 df/dx = 4𝑥3 − 4𝑥 + 4𝑦 at ( -2 , 2 ) df/dy = 4𝑦3 + 4𝑥 − 4𝑦 For Stationary points We Have df/dx = 0 ,df/dy = 0 , so… 𝑥3 − 𝑥 + 𝑦 = 0 𝑦3 + 𝑥 − 𝑦 = 0 𝑥3 + 𝑦3 = 0 𝑥 + 𝑦 𝑥2 − 𝑥𝑦 + 𝑦2 = 0 𝑥 + 𝑦 = 0 𝑦 = −𝑥 Putting 𝑦 = −𝑥 𝑥3 − 2𝑥 = 0 𝑥 𝑥2 − 2 = 0 (√2 , -√2) , (-√2 , √2) , ( 0,0) are Stationary Points r = 12𝑥2 − 4 ,s = 4 ,t = 12𝑦2 − 4 ,t = 12𝑦2 − 4 rt-𝑠2 = 144𝑥2𝑦2 − 48𝑥2 − 48𝑦2 At (√2 ,-√2) rt-𝑠2 = 384 > 0 r = 20>0 (√2 , √2) is a Minimum Point rt-𝑠2 = 384 > 0 r = 12𝑥2 − 4 = 20 > 20 so ( -2 ,2) is inimum Point at (0 , 0) rt-𝑠2 = 0 At (√2 ,-√2) rt-𝑠2 = 384 > 0 r = 20>0 (√2 , √2) is a Minimum Point rt-𝑠2 = 384 > 0 r = 12𝑥2 − 4 = 20 > 20 so ( -2 ,2) is inimum Point at (0 , 0) rt-𝑠2 = 0
- 6. • Ex.2 Find The Extrime Value of sinx siny sin (x+y) • df/dx = cosx siny sin (x+y) +sinx siny cos (x+y) • =siny sin (2x+y) • df/dy = sinx cosy sin (x+y +sinx siny cos (x+y) • =sinx sin (x+2y) • For Stationary Points we Have • Df/dx = 0 ,df/dy = 0 • =siny [cosx (sin (x+y)]+sinx [cos (x+y )] = 0 • Tan (x+y) = -tanx ; x≠π/2 ,3π/2 ,x+y≠π/2 ,3π/2 • We have tanx = tany ;x = y • Tan2x = -tanx = sin2x/cos2x = -sinx/cosx • sin3x/cos2x cosx = 0 • sin3x = 0 • 3x = π • r = 𝑑2 𝑓 𝑑𝑥2 = 2𝑦 cos (2𝑥 + 𝑦) • S = 𝑑2𝑓 𝑑𝑥𝑑𝑦 = 𝑐𝑜𝑠𝑥 sin 𝑥 + 2𝑦 + 𝑥 cos (𝑥 + 2𝑦) • T = 𝑓 𝑑𝑦2 = 2𝑠𝑖𝑛𝑥cos (x+2y) • Thus, f(x,y) is ,aximum at (π/3 , π/3) and the Maximum Value is • F (π/3 , π/3) = sin π/3 sin π/3 sin 2π/3 • = (√3/2) (√3/2) (√3/2) • = 3√3/8
- 7. THANK YOU