rev2023.3.3.43278. To find local maximum or minimum, first, the first derivative of the function needs to be found. And, in second-order derivative test we check the sign of the second-order derivatives at critical points to find the points of local maximum and minimum. You divide this number line into four regions: to the left of 2, from 2 to 0, from 0 to 2, and to the right of 2. Apply the distributive property. \end{align}. Tap for more steps. How to Find Local Extrema with the First Derivative Test Find all critical numbers c of the function f ( x) on the open interval ( a, b). If b2 - 3ac 0, then the cubic function has a local maximum and a local minimum. How to find local maxima of a function | Math Assignments Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. How to find the local maximum of a cubic function Maxima and Minima from Calculus. If the function goes from increasing to decreasing, then that point is a local maximum. If the first element x [1] is the global maximum, it is ignored, because there is no information about the previous emlement. The smallest value is the absolute minimum, and the largest value is the absolute maximum. Well think about what happens if we do what you are suggesting. That is, find f ( a) and f ( b). How to find relative extrema with second derivative test Identify those arcade games from a 1983 Brazilian music video, How to tell which packages are held back due to phased updates, How do you get out of a corner when plotting yourself into a corner. So it works out the values in the shifts of the maxima or minima at (0,0) , in the specific quadratic, to deduce the actual maxima or minima in any quadratic. Dummies helps everyone be more knowledgeable and confident in applying what they know. 0 = y &= ax^2 + bx + c \\ &= at^2 + c - \frac{b^2}{4a}. FindMaximum [f, {x, x 0, x 1}] searches for a local maximum in f using x 0 and x 1 as the first two values of x, avoiding the use of derivatives. Maximum and minimum - WikipediaHow to find the maximum of a function calculus - Math Tutor And there is an important technical point: The function must be differentiable (the derivative must exist at each point in its domain). In either case, talking about tangent lines at these maximum points doesn't really make sense, does it? Therefore, first we find the difference. It's good practice for thinking clearly, and it can also help to understand those times when intuition differs from reality. And that first derivative test will give you the value of local maxima and minima. AP Calculus Review: Finding Absolute Extrema - MagooshHow to find local maximum | Math Assignments For example, suppose we want to find the following function's global maximum and global minimum values on the indicated interval. The vertex of $y = A(x - k)^2 + j$ is just shifted up $j$, so it is $(k, j)$. [closed], meta.math.stackexchange.com/questions/5020/, We've added a "Necessary cookies only" option to the cookie consent popup. The function must also be continuous, but any function that is differentiable is also continuous, so we are covered. or the minimum value of a quadratic equation. Get support from expert teachers If you're looking for expert teachers to help support your learning, look no further than our online tutoring services. Follow edited Feb 12, 2017 at 10:11. How do you find a local minimum of a graph using. Set the derivative equal to zero and solve for x. the point is an inflection point). Not all functions have a (local) minimum/maximum. \end{align} In calculus, a derivative test uses the derivatives of a function to locate the critical points of a function and determine whether each point is a local maximum, a local minimum, or a saddle point.Derivative tests can also give information about the concavity of a function.. Step 1: Differentiate the given function. Using the assumption that the curve is symmetric around a vertical axis, Finding Maxima/Minima of Polynomials without calculus? This is one of the best answer I have come across, Yes a variation of this idea can be used to find the minimum too. How can I know whether the point is a maximum or minimum without much calculation? @return returns the indicies of local maxima. Then using the plot of the function, you can determine whether the points you find were a local minimum or a local maximum. Global Extrema - S.O.S. Math To find a local max and min value of a function, take the first derivative and set it to zero. TI-84 Plus Lesson - Module 13.1: Critical Points | TI - Texas Instruments The partial derivatives will be 0. A low point is called a minimum (plural minima). \tag 1 Often, they are saddle points. &= \pm \frac{\sqrt{b^2 - 4ac}}{\lvert 2a \rvert}\\ Find all the x values for which f'(x) = 0 and list them down. It very much depends on the nature of your signal. Direct link to Alex Sloan's post Well think about what hap, Posted 5 years ago. On the contrary, the equation $y = at^2 + c - \dfrac{b^2}{4a}$ tells us that In general, local maxima and minima of a function f f are studied by looking for input values a a where f' (a) = 0 f (a) = 0. Maxima and Minima are one of the most common concepts in differential calculus. The local min is (3,3) and the local max is (5,1) with an inflection point at (4,2). They are found by setting derivative of the cubic equation equal to zero obtaining: f (x) = 3ax2 + 2bx + c = 0. It's obvious this is true when $b = 0$, and if we have plotted 1.If f(x) is a continuous function in its domain, then at least one maximum or one minimum should lie between equal values of f(x). we may observe enough appearance of symmetry to suppose that it might be true in general. 1. Solve the system of equations to find the solutions for the variables. Its increasing where the derivative is positive, and decreasing where the derivative is negative. I think this is a good answer to the question I asked. So say the function f'(x) is 0 at the points x1,x2 and x3. Given a differentiable function, the first derivative test can be applied to determine any local maxima or minima of the given function through the steps given below. Maxima and Minima: Local and Absolute Maxima and Minima - Embibe Dummies has always stood for taking on complex concepts and making them easy to understand. Now, heres the rocket science. First Derivative Test for Local Maxima and Local Minima. &= c - \frac{b^2}{4a}. Heres how:\r\n\r\n \t
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Take a number line and put down the critical numbers you have found: 0, 2, and 2.
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You divide this number line into four regions: to the left of 2, from 2 to 0, from 0 to 2, and to the right of 2.
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Pick a value from each region, plug it into the first derivative, and note whether your result is positive or negative.
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For this example, you can use the numbers 3, 1, 1, and 3 to test the regions.
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These four results are, respectively, positive, negative, negative, and positive.
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Take your number line, mark each region with the appropriate positive or negative sign, and indicate where the function is increasing and decreasing.
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Its increasing where the derivative is positive, and decreasing where the derivative is negative. Math: How to Find the Minimum and Maximum of a Function Example. Our book does this with the use of graphing calculators, but I was wondering if there is a way to find the critical points without derivatives. Find the global minimum of a function of two variables without derivatives. In the last slide we saw that. \begin{align} get the first and the second derivatives find zeros of the first derivative (solve quadratic equation) check the second derivative in found \begin{equation} f(x)=3 x^{2}-18 x+5,[0,7] \end{equation} if we make the substitution $x = -\dfrac b{2a} + t$, that means that the curve $y = ax^2 + bx + c$ is symmetric around a vertical axis. We try to find a point which has zero gradients . Learn more about Stack Overflow the company, and our products. Finding Maxima and Minima using Derivatives - mathsisfun.com Any such value can be expressed by its difference This test is based on the Nobel-prize-caliber ideas that as you go over the top of a hill, first you go up and then you go down, and that when you drive into and out of a valley, you go down and then up. Find the local maximum and local minimum values by using 1st derivative test for the function, f (x) = 3x4+4x3 -12x2+12. The function switches from increasing to decreasing at 2; in other words, you go up to 2 and then down. For example. The main purpose for determining critical points is to locate relative maxima and minima, as in single-variable calculus. This calculus stuff is pretty amazing, eh?\r\n\r\n\r\n\r\nThe figure shows the graph of\r\n\r\n\r\n\r\nTo find the critical numbers of this function, heres what you do:\r\n
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Find the first derivative of f using the power rule.
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Set the derivative equal to zero and solve for x.
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x = 0, 2, or 2.
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These three x-values are the critical numbers of f.Additional critical numbers could exist if the first derivative were undefined at some x-values, but because the derivative
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is defined for all input values, the above solution set, 0, 2, and 2, is the complete list of critical numbers. When the second derivative is negative at x=c, then f(c) is maximum.Feb 21, 2022 14.7 Maxima and minima - Whitman College So we can't use the derivative method for the absolute value function. 3) f(c) is a local . algebra-precalculus; Share. We say local maximum (or minimum) when there may be higher (or lower) points elsewhere but not nearby. Extended Keyboard. To find a local max or min we essentially want to find when the difference between the values in the list (3-1, 9-3.) At this point the tangent has zero slope.The graph has a local minimum at the point where the graph changes from decreasing to increasing. Extrema (Local and Absolute) | Brilliant Math & Science Wiki Homework Support Solutions. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. It is an Inflection Point ("saddle point") the slope does become zero, but it is neither a maximum nor minimum. $y = ax^2 + bx + c$ for various other values of $a$, $b$, and $c$, Whether it's to pass that big test, qualify for that big promotion or even master that cooking technique; people who rely on dummies, rely on it to learn the critical skills and relevant information necessary for success. For these values, the function f gets maximum and minimum values. So, at 2, you have a hill or a local maximum. Step 1. f ' (x) = 0, Set derivative equal to zero and solve for "x" to find critical points. How to find the maximum and minimum of a multivariable function? &= at^2 + c - \frac{b^2}{4a}. Direct link to sprincejindal's post When talking about Saddle, Posted 7 years ago. Intuitively, it is a special point in the input space where taking a small step in any direction can only decrease the value of the function. Finding Maxima and Minima using Derivatives f(x) be a real function of a real variable defined in (a,b) and differentiable in the point x0(a,b) x0 to be a local minimum or maximum is . Direct link to Raymond Muller's post Nope. the graph of its derivative f '(x) passes through the x axis (is equal to zero). Connect and share knowledge within a single location that is structured and easy to search. \"https://sb\" : \"http://b\") + \".scorecardresearch.com/beacon.js\";el.parentNode.insertBefore(s, el);})();\r\n","enabled":true},{"pages":["all"],"location":"footer","script":"\r\n
\r\n","enabled":false},{"pages":["all"],"location":"header","script":"\r\n","enabled":false},{"pages":["article"],"location":"header","script":" ","enabled":true},{"pages":["homepage"],"location":"header","script":"","enabled":true},{"pages":["homepage","article","category","search"],"location":"footer","script":"\r\n\r\n","enabled":true}]}},"pageScriptsLoadedStatus":"success"},"navigationState":{"navigationCollections":[{"collectionId":287568,"title":"BYOB (Be Your Own Boss)","hasSubCategories":false,"url":"/collection/for-the-entry-level-entrepreneur-287568"},{"collectionId":293237,"title":"Be a Rad Dad","hasSubCategories":false,"url":"/collection/be-the-best-dad-293237"},{"collectionId":295890,"title":"Career Shifting","hasSubCategories":false,"url":"/collection/career-shifting-295890"},{"collectionId":294090,"title":"Contemplating the Cosmos","hasSubCategories":false,"url":"/collection/theres-something-about-space-294090"},{"collectionId":287563,"title":"For Those Seeking Peace of Mind","hasSubCategories":false,"url":"/collection/for-those-seeking-peace-of-mind-287563"},{"collectionId":287570,"title":"For the Aspiring Aficionado","hasSubCategories":false,"url":"/collection/for-the-bougielicious-287570"},{"collectionId":291903,"title":"For the Budding Cannabis Enthusiast","hasSubCategories":false,"url":"/collection/for-the-budding-cannabis-enthusiast-291903"},{"collectionId":291934,"title":"For the Exam-Season Crammer","hasSubCategories":false,"url":"/collection/for-the-exam-season-crammer-291934"},{"collectionId":287569,"title":"For the Hopeless Romantic","hasSubCategories":false,"url":"/collection/for-the-hopeless-romantic-287569"},{"collectionId":296450,"title":"For the Spring Term Learner","hasSubCategories":false,"url":"/collection/for-the-spring-term-student-296450"}],"navigationCollectionsLoadedStatus":"success","navigationCategories":{"books":{"0":{"data":[{"categoryId":33512,"title":"Technology","hasSubCategories":true,"url":"/category/books/technology-33512"},{"categoryId":33662,"title":"Academics & The Arts","hasSubCategories":true,"url":"/category/books/academics-the-arts-33662"},{"categoryId":33809,"title":"Home, Auto, & Hobbies","hasSubCategories":true,"url":"/category/books/home-auto-hobbies-33809"},{"categoryId":34038,"title":"Body, Mind, & Spirit","hasSubCategories":true,"url":"/category/books/body-mind-spirit-34038"},{"categoryId":34224,"title":"Business, Careers, & Money","hasSubCategories":true,"url":"/category/books/business-careers-money-34224"}],"breadcrumbs":[],"categoryTitle":"Level 0 Category","mainCategoryUrl":"/category/books/level-0-category-0"}},"articles":{"0":{"data":[{"categoryId":33512,"title":"Technology","hasSubCategories":true,"url":"/category/articles/technology-33512"},{"categoryId":33662,"title":"Academics & The Arts","hasSubCategories":true,"url":"/category/articles/academics-the-arts-33662"},{"categoryId":33809,"title":"Home, Auto, & Hobbies","hasSubCategories":true,"url":"/category/articles/home-auto-hobbies-33809"},{"categoryId":34038,"title":"Body, Mind, & Spirit","hasSubCategories":true,"url":"/category/articles/body-mind-spirit-34038"},{"categoryId":34224,"title":"Business, Careers, & Money","hasSubCategories":true,"url":"/category/articles/business-careers-money-34224"}],"breadcrumbs":[],"categoryTitle":"Level 0 Category","mainCategoryUrl":"/category/articles/level-0-category-0"}}},"navigationCategoriesLoadedStatus":"success"},"searchState":{"searchList":[],"searchStatus":"initial","relatedArticlesList":[],"relatedArticlesStatus":"initial"},"routeState":{"name":"Article3","path":"/article/academics-the-arts/math/pre-calculus/how-to-find-local-extrema-with-the-first-derivative-test-192147/","hash":"","query":{},"params":{"category1":"academics-the-arts","category2":"math","category3":"pre-calculus","article":"how-to-find-local-extrema-with-the-first-derivative-test-192147"},"fullPath":"/article/academics-the-arts/math/pre-calculus/how-to-find-local-extrema-with-the-first-derivative-test-192147/","meta":{"routeType":"article","breadcrumbInfo":{"suffix":"Articles","baseRoute":"/category/articles"},"prerenderWithAsyncData":true},"from":{"name":null,"path":"/","hash":"","query":{},"params":{},"fullPath":"/","meta":{}}},"dropsState":{"submitEmailResponse":false,"status":"initial"},"sfmcState":{"status":"initial"},"profileState":{"auth":{},"userOptions":{},"status":"success"}}, The Differences between Pre-Calculus and Calculus, Pre-Calculus: 10 Habits to Adjust before Calculus. Here, we'll focus on finding the local minimum. Here's a video of this graph rotating in space: Well, mathematicians thought so, and they had one of those rare moments of deciding on a good name for something: "so it's not enough for the gradient to be, I'm glad you asked! Best way to find local minimum and maximum (where derivatives = 0 what R should be? Take the derivative of the slope (the second derivative of the original function): This means the slope is continually getting smaller (10): traveling from left to right the slope starts out positive (the function rises), goes through zero (the flat point), and then the slope becomes negative (the function falls): A slope that gets smaller (and goes though 0) means a maximum. 5.1 Maxima and Minima. $ax^2 + bx + c = at^2 + c - \dfrac{b^2}{4a}$ There is only one equation with two unknown variables. Or if $x > |b|/2$ then $(x+ h)^2 + b(x + h) = x^2 + bx +h(2x + b) + h^2 > 0$ so the expression has no max value. So if $ax^2 + bx + c = a(x^2 + x b/a)+c := a(x^2 + b'x) + c$ So finding the max/min is simply a matter of finding the max/min of $x^2 + b'x$ and multiplying by $a$ and adding $c$. f(x)f(x0) why it is allowed to be greater or EQUAL ? And because the sign of the first derivative doesnt switch at zero, theres neither a min nor a max at that x-value.\r\n
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Obtain the function values (in other words, the heights) of these two local extrema by plugging the x-values into the original function.
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Thus, the local max is located at (2, 64), and the local min is at (2, 64). FindMaximumWolfram Language Documentation How to find the local maximum and minimum of a cubic function. Why are non-Western countries siding with China in the UN? Find the partial derivatives. Finding the local minimum using derivatives. How to Find Local Extrema with the Second Derivative Test So x = -2 is a local maximum, and x = 8 is a local minimum. The function f(x)=sin(x) has an inflection point at x=0, but the derivative is not 0 there. How to find local maximum of cubic function. The largest value found in steps 2 and 3 above will be the absolute maximum and the . \end{align} The function switches from increasing to decreasing at 2; in other words, you go up to 2 and then down. The roots of the equation @Karlie Kloss Technically speaking this solution is also not without completion of squares because you are still using the quadratic formula and how do you get that??? Using the second-derivative test to determine local maxima and minima. $\left(-\frac ba, c\right)$ and $(0, c)$, that is, it is If f ( x) < 0 for all x I, then f is decreasing on I . The question then is, what is the proof of the quadratic formula that does not use any form of completing the square? Finding maxima and minima using derivatives - BYJUS I suppose that would depend on the specific function you were looking at at the time, and the context might make it clear. You then use the First Derivative Test. This is because the values of x 2 keep getting larger and larger without bound as x . if this is just an inspired guess) That's a bit of a mouthful, so let's break it down: We can then translate this definition from math-speak to something more closely resembling English as follows: Posted 7 years ago. Bulk update symbol size units from mm to map units in rule-based symbology.
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\r\n\r\nThe figure shows the graph of\r\n\r\n![\"image1.png\"](\"https://www.dummies.com/wp-content/uploads/204564.image1.png\")
\r\n\r\nTo find the critical numbers of this function, heres what you do:\r\n![\"image2.png\"](\"https://www.dummies.com/wp-content/uploads/204565.image2.png\")
\r\n \t![\"image3.png\"](\"https://www.dummies.com/wp-content/uploads/204566.image3.png\")
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\r\n","enabled":true}]}},"pageScriptsLoadedStatus":"success"},"navigationState":{"navigationCollections":[{"collectionId":287568,"title":"BYOB (Be Your Own Boss)","hasSubCategories":false,"url":"/collection/for-the-entry-level-entrepreneur-287568"},{"collectionId":293237,"title":"Be a Rad Dad","hasSubCategories":false,"url":"/collection/be-the-best-dad-293237"},{"collectionId":295890,"title":"Career Shifting","hasSubCategories":false,"url":"/collection/career-shifting-295890"},{"collectionId":294090,"title":"Contemplating the Cosmos","hasSubCategories":false,"url":"/collection/theres-something-about-space-294090"},{"collectionId":287563,"title":"For Those Seeking Peace of Mind","hasSubCategories":false,"url":"/collection/for-those-seeking-peace-of-mind-287563"},{"collectionId":287570,"title":"For the Aspiring Aficionado","hasSubCategories":false,"url":"/collection/for-the-bougielicious-287570"},{"collectionId":291903,"title":"For the Budding Cannabis Enthusiast","hasSubCategories":false,"url":"/collection/for-the-budding-cannabis-enthusiast-291903"},{"collectionId":291934,"title":"For the Exam-Season Crammer","hasSubCategories":false,"url":"/collection/for-the-exam-season-crammer-291934"},{"collectionId":287569,"title":"For the Hopeless Romantic","hasSubCategories":false,"url":"/collection/for-the-hopeless-romantic-287569"},{"collectionId":296450,"title":"For the Spring Term Learner","hasSubCategories":false,"url":"/collection/for-the-spring-term-student-296450"}],"navigationCollectionsLoadedStatus":"success","navigationCategories":{"books":{"0":{"data":[{"categoryId":33512,"title":"Technology","hasSubCategories":true,"url":"/category/books/technology-33512"},{"categoryId":33662,"title":"Academics & The Arts","hasSubCategories":true,"url":"/category/books/academics-the-arts-33662"},{"categoryId":33809,"title":"Home, Auto, & Hobbies","hasSubCategories":true,"url":"/category/books/home-auto-hobbies-33809"},{"categoryId":34038,"title":"Body, Mind, & Spirit","hasSubCategories":true,"url":"/category/books/body-mind-spirit-34038"},{"categoryId":34224,"title":"Business, Careers, & Money","hasSubCategories":true,"url":"/category/books/business-careers-money-34224"}],"breadcrumbs":[],"categoryTitle":"Level 0 Category","mainCategoryUrl":"/category/books/level-0-category-0"}},"articles":{"0":{"data":[{"categoryId":33512,"title":"Technology","hasSubCategories":true,"url":"/category/articles/technology-33512"},{"categoryId":33662,"title":"Academics & The Arts","hasSubCategories":true,"url":"/category/articles/academics-the-arts-33662"},{"categoryId":33809,"title":"Home, Auto, & Hobbies","hasSubCategories":true,"url":"/category/articles/home-auto-hobbies-33809"},{"categoryId":34038,"title":"Body, Mind, & Spirit","hasSubCategories":true,"url":"/category/articles/body-mind-spirit-34038"},{"categoryId":34224,"title":"Business, Careers, & Money","hasSubCategories":true,"url":"/category/articles/business-careers-money-34224"}],"breadcrumbs":[],"categoryTitle":"Level 0 Category","mainCategoryUrl":"/category/articles/level-0-category-0"}}},"navigationCategoriesLoadedStatus":"success"},"searchState":{"searchList":[],"searchStatus":"initial","relatedArticlesList":[],"relatedArticlesStatus":"initial"},"routeState":{"name":"Article3","path":"/article/academics-the-arts/math/pre-calculus/how-to-find-local-extrema-with-the-first-derivative-test-192147/","hash":"","query":{},"params":{"category1":"academics-the-arts","category2":"math","category3":"pre-calculus","article":"how-to-find-local-extrema-with-the-first-derivative-test-192147"},"fullPath":"/article/academics-the-arts/math/pre-calculus/how-to-find-local-extrema-with-the-first-derivative-test-192147/","meta":{"routeType":"article","breadcrumbInfo":{"suffix":"Articles","baseRoute":"/category/articles"},"prerenderWithAsyncData":true},"from":{"name":null,"path":"/","hash":"","query":{},"params":{},"fullPath":"/","meta":{}}},"dropsState":{"submitEmailResponse":false,"status":"initial"},"sfmcState":{"status":"initial"},"profileState":{"auth":{},"userOptions":{},"status":"success"}}, The Differences between Pre-Calculus and Calculus, Pre-Calculus: 10 Habits to Adjust before Calculus. Here, we'll focus on finding the local minimum. Here's a video of this graph rotating in space: Well, mathematicians thought so, and they had one of those rare moments of deciding on a good name for something: "so it's not enough for the gradient to be, I'm glad you asked! Best way to find local minimum and maximum (where derivatives = 0 what R should be? Take the derivative of the slope (the second derivative of the original function): This means the slope is continually getting smaller (10): traveling from left to right the slope starts out positive (the function rises), goes through zero (the flat point), and then the slope becomes negative (the function falls): A slope that gets smaller (and goes though 0) means a maximum. 5.1 Maxima and Minima. $ax^2 + bx + c = at^2 + c - \dfrac{b^2}{4a}$ There is only one equation with two unknown variables. Or if $x > |b|/2$ then $(x+ h)^2 + b(x + h) = x^2 + bx +h(2x + b) + h^2 > 0$ so the expression has no max value. So if $ax^2 + bx + c = a(x^2 + x b/a)+c := a(x^2 + b'x) + c$ So finding the max/min is simply a matter of finding the max/min of $x^2 + b'x$ and multiplying by $a$ and adding $c$. f(x)f(x0) why it is allowed to be greater or EQUAL ? And because the sign of the first derivative doesnt switch at zero, theres neither a min nor a max at that x-value.\r\n
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