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! Can airtags be tracked from an iMac desktop, with no iPhone? This is almost the same as completing the square but .. for giggles. She taught at Bradley University in Peoria, Illinois for more than 30 years, teaching algebra, business calculus, geometry, and finite mathematics. Multiply that out, you get $y = Ax^2 - 2Akx + Ak^2 + j$. In particular, we want to differentiate between two types of minimum or . 2.) Step 1. f ' (x) = 0, Set derivative equal to zero and solve for "x" to find critical points. A little algebra (isolate the $at^2$ term on one side and divide by $a$) Why is there a voltage on my HDMI and coaxial cables? The graph of a function y = f(x) has a local maximum at the point where the graph changes from increasing to decreasing. Where the slope is zero. Check 452+ Teachers 78% Recurring customers 99497 Clients Get Homework Help 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. t &= \pm \sqrt{\frac{b^2}{4a^2} - \frac ca} \\ Direct link to Andrea Menozzi's post what R should be? Solve (1) for $k$ and plug it into (2), then solve for $j$,you get: $$k = \frac{-b}{2a}$$ If there is a multivariable function and we want to find its maximum point, we have to take the partial derivative of the function with respect to both the variables. Math can be tough to wrap your head around, but with a little practice, it can be a breeze! Find all the x values for which f'(x) = 0 and list them down. \begin{align} Given a function f f and interval [a, \, b] [a . Dont forget, though, that not all critical points are necessarily local extrema.\r\n\r\nThe first step in finding a functions local extrema is to find its critical numbers (the x-values of the critical points). The function f ( x) = 3 x 4 4 x 3 12 x 2 + 3 has first derivative. \begin{align} Determine math problem In order to determine what the math problem is, you will need to look at the given information and find the key details. Global Maximum (Absolute Maximum): Definition. Follow edited Feb 12, 2017 at 10:11. @param x numeric vector. A high point is called a maximum (plural maxima). neither positive nor negative (i.e. Evaluating derivative with respect to x. f' (x) = d/dx [3x4+4x3 -12x2+12] Since the function involves power functions, so by using power rule of derivative, A derivative basically finds the slope of a function. Because the derivative (and the slope) of f equals zero at these three critical numbers, the curve has horizontal tangents at these numbers.

\r\n\r\n\r\nNow that youve got the list of critical numbers, you need to determine whether peaks or valleys or neither occur at those x-values. 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.) If the second derivative is greater than zerof(x1)0 f ( x 1 ) 0 , then the limiting point (x1) ( x 1 ) is the local minima. Find all critical numbers c of the function f ( x) on the open interval ( a, b). Direct link to shivnaren's post _In machine learning and , Posted a year ago. To find the local maximum and minimum values of the function, set the derivative equal to and solve. Do my homework for me. The vertex of $y = A(x - k)^2 + j$ is just shifted up $j$, so it is $(k, j)$. If b2 - 3ac 0, then the cubic function has a local maximum and a local minimum. So say the function f'(x) is 0 at the points x1,x2 and x3. 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. if this is just an inspired guess) I have a "Subject:, Posted 5 years ago. iii. \begin{align} Many of our applications in this chapter will revolve around minimum and maximum values of a function. Thus, the local max is located at (2, 64), and the local min is at (2, 64). She taught at Bradley University in Peoria, Illinois for more than 30 years, teaching algebra, business calculus, geometry, and finite mathematics. This is called the Second Derivative Test. Maxima and Minima are one of the most common concepts in differential calculus. 18B Local Extrema 2 Definition Let S be the domain of f such that c is an element of S. Then, 1) f(c) is a local maximum value of f if there exists an interval (a,b) containing c such that f(c) is the maximum value of f on (a,b)S. 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\r\n \t

\r\n

Obtain the function values (in other words, the heights) of these two local extrema by plugging the x-values into the original function.

\r\n\r\n

Thus, the local max is located at (2, 64), and the local min is at (2, 64). It very much depends on the nature of your signal. ","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/8985"}}],"primaryCategoryTaxonomy":{"categoryId":33727,"title":"Pre-Calculus","slug":"pre-calculus","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33727"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[{"articleId":260218,"title":"Special Function Types and Their Graphs","slug":"special-function-types-and-their-graphs","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/260218"}},{"articleId":260215,"title":"The Differences between Pre-Calculus and Calculus","slug":"the-differences-between-pre-calculus-and-calculus","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/260215"}},{"articleId":260207,"title":"10 Polar Graphs","slug":"10-polar-graphs","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/260207"}},{"articleId":260183,"title":"Pre-Calculus: 10 Habits to Adjust before Calculus","slug":"pre-calculus-10-habits-to-adjust-before-calculus","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/260183"}},{"articleId":208308,"title":"Pre-Calculus For Dummies Cheat Sheet","slug":"pre-calculus-for-dummies-cheat-sheet","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/208308"}}],"fromCategory":[{"articleId":262884,"title":"10 Pre-Calculus Missteps to Avoid","slug":"10-pre-calculus-missteps-to-avoid","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262884"}},{"articleId":262851,"title":"Pre-Calculus Review of Real Numbers","slug":"pre-calculus-review-of-real-numbers","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262851"}},{"articleId":262837,"title":"Fundamentals of Pre-Calculus","slug":"fundamentals-of-pre-calculus","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262837"}},{"articleId":262652,"title":"Complex Numbers and Polar Coordinates","slug":"complex-numbers-and-polar-coordinates","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262652"}},{"articleId":260218,"title":"Special Function Types and Their Graphs","slug":"special-function-types-and-their-graphs","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/260218"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":282496,"slug":"pre-calculus-for-dummies-3rd-edition","isbn":"9781119508779","categoryList":["academics-the-arts","math","pre-calculus"],"amazon":{"default":"https://www.amazon.com/gp/product/1119508770/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"https://www.amazon.ca/gp/product/1119508770/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"http://www.tkqlhce.com/click-9208661-13710633?url=https://www.chapters.indigo.ca/en-ca/books/product/1119508770-item.html&cjsku=978111945484","gb":"https://www.amazon.co.uk/gp/product/1119508770/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"https://www.amazon.de/gp/product/1119508770/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"https://www.dummies.com/wp-content/uploads/pre-calculus-for-dummies-3rd-edition-cover-9781119508779-203x255.jpg","width":203,"height":255},"title":"Pre-Calculus For Dummies","testBankPinActivationLink":"","bookOutOfPrint":false,"authorsInfo":"

Mary Jane Sterling aught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois for more than 30 years. Direct link to kashmalahassan015's post questions of triple deriv, Posted 7 years ago. Because the derivative (and the slope) of f equals zero at these three critical numbers, the curve has horizontal tangents at these numbers.

\r\n

\r\n\r\nNow that youve got the list of critical numbers, you need to determine whether peaks or valleys or neither occur at those x-values. ", When talking about Saddle point in this article. The main purpose for determining critical points is to locate relative maxima and minima, as in single-variable calculus. It says 'The single-variable function f(x) = x^2 has a local minimum at x=0, and. it is less than 0, so 3/5 is a local maximum, it is greater than 0, so +1/3 is a local minimum, equal to 0, then the test fails (there may be other ways of finding out though). Direct link to Andrea Menozzi's post f(x)f(x0) why it is allo, Posted 3 years ago. So the vertex occurs at $(j, k) = \left(\frac{-b}{2a}, \frac{4ac - b^2}{4a}\right)$. Let $y := x - b'/2$ then $x(x + b')=(y -b'/2)(y + b'/2)= y^2 - (b'^2/4)$. The maximum or minimum over the entire function is called an "Absolute" or "Global" maximum or minimum. Second Derivative Test. In particular, I show students how to make a sign ch. Hence if $(x,c)$ is on the curve, then either $ax + b = 0$ or $x = 0$. \end{align} Why can ALL quadratic equations be solved by the quadratic formula? These four results are, respectively, positive, negative, negative, and positive. 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. The function switches from increasing to decreasing at 2; in other words, you go up to 2 and then down. 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. Perhaps you find yourself running a company, and you've come up with some function to model how much money you can expect to make based on a number of parameters, such as employee salaries, cost of raw materials, etc., and you want to find the right combination of resources that will maximize your revenues. The first step in finding a functions local extrema is to find its critical numbers (the x-values of the critical points). A point where the derivative of the function is zero but the derivative does not change sign is known as a point of inflection , or saddle point . Glitch? local minimum calculator. Finding Extreme Values of a Function Theorem 2 says that if a function has a first derivative at an interior point where there is a local extremum, then the derivative must equal zero at that . I think what you mean to say is simply that a function's derivative can equal 0 at a point without having an extremum at that point, which is related to the fact that the second derivative at that point is 0, i.e. Dont forget, though, that not all critical points are necessarily local extrema.\r\n\r\nThe first step in finding a functions local extrema is to find its critical numbers (the x-values of the critical points). So thank you to the creaters of This app, a best app, awesome experience really good app with every feature I ever needed in a graphic calculator without needind to pay, some improvements to be made are hand writing recognition, and also should have a writing board for faster calculations, needs a dark mode too. Using the assumption that the curve is symmetric around a vertical axis, Direct link to zk306950's post Is the following true whe, Posted 5 years ago. The word "critical" always seemed a bit over dramatic to me, as if the function is about to die near those points. any val, Posted 3 years ago. It is an Inflection Point ("saddle point") the slope does become zero, but it is neither a maximum nor minimum. We assume (for the sake of discovery; for this purpose it is good enough Homework Support Solutions. So now you have f'(x). and therefore $y_0 = c - \dfrac{b^2}{4a}$ is a minimum. Conversely, because the function switches from decreasing to increasing at 2, you have a valley there or a local minimum. $-\dfrac b{2a}$. . If there is a plateau, the first edge is detected. Similarly, if the graph has an inverted peak at a point, we say the function has a, Tangent lines at local extrema have slope 0. Note: all turning points are stationary points, but not all stationary points are turning points. Note that the proof made no assumption about the symmetry of the curve. Instead, the quantity $c - \dfrac{b^2}{4a}$ just "appeared" in the Tap for more steps. To find a local max and min value of a function, take the first derivative and set it to zero. In other words . wolog $a = 1$ and $c = 0$. Finding sufficient conditions for maximum local, minimum local and saddle point. for every point $(x,y)$ on the curve such that $x \neq x_0$, isn't it just greater? &= \pm \frac{\sqrt{b^2 - 4ac}}{2a}, If b2 - 3ac 0, then the cubic function has a local maximum and a local minimum. Browse other questions tagged, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site. [closed], meta.math.stackexchange.com/questions/5020/, We've added a "Necessary cookies only" option to the cookie consent popup. Often, they are saddle points. This tells you that f is concave down where x equals -2, and therefore that there's a local max y &= c. \\ 1. This calculus stuff is pretty amazing, eh? Max and Min of Functions without Derivative I was curious to know if there is a general way to find the max and min of cubic functions without using derivatives. One approach for finding the maximum value of $y$ for $y=ax^2+bx+c$ would be to see how large $y$ can be before the equation has no solution for $x$. 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. by taking the second derivative), you can get to it by doing just that. We cant have the point x = x0 then yet when we say for all x we mean for the entire domain of the function. It's good practice for thinking clearly, and it can also help to understand those times when intuition differs from reality. Explanation: To find extreme values of a function f, set f '(x) = 0 and solve. 2. 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. Setting $x_1 = -\dfrac ba$ and $x_2 = 0$, we can plug in these two values You then use the First Derivative Test. \end{align} When a function's slope is zero at x, and the second derivative at x is: less than 0, it is a local maximum; greater than 0, it is a local minimum; equal to 0, then the test fails (there may be other ways of finding out though) Therefore, first we find the difference. But there is also an entirely new possibility, unique to multivariable functions. The Global Minimum is Infinity. People often write this more compactly like this: The thinking behind the words "stable" and "stationary" is that when you move around slightly near this input, the value of the function doesn't change significantly. 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. Worked Out Example. i am trying to find out maximum and minimum value of above questions without using derivative but not be able to evaluate , could some help me. How to find the local maximum and minimum of a cubic function. from $-\dfrac b{2a}$, that is, we let All in all, we can say that the steps to finding the maxima/minima/saddle point (s) of a multivariable function are: 1.) Find the minimum of $\sqrt{\cos x+3}+\sqrt{2\sin x+7}$ without derivative. Do new devs get fired if they can't solve a certain bug? 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). All local extrema are critical points. If you're seeing this message, it means we're having trouble loading external resources on our website. . It's not true. In defining a local maximum, let's use vector notation for our input, writing it as. {"appState":{"pageLoadApiCallsStatus":true},"articleState":{"article":{"headers":{"creationTime":"2016-03-26T21:18:56+00:00","modifiedTime":"2021-07-09T18:46:09+00:00","timestamp":"2022-09-14T18:18:24+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Math","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33720"},"slug":"math","categoryId":33720},{"name":"Pre-Calculus","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33727"},"slug":"pre-calculus","categoryId":33727}],"title":"How to Find Local Extrema with the First Derivative Test","strippedTitle":"how to find local extrema with the first derivative test","slug":"how-to-find-local-extrema-with-the-first-derivative-test","canonicalUrl":"","seo":{"metaDescription":"All local maximums and minimums on a function's graph called local extrema occur at critical points of the function (where the derivative is zero or undefin","noIndex":0,"noFollow":0},"content":"All local maximums and minimums on a function's graph called local extrema occur at critical points of the function (where the derivative is zero or undefined). Where does it flatten out? This is like asking how to win a martial arts tournament while unconscious. Finding sufficient conditions for maximum local, minimum local and . There are multiple ways to do so. Using the second-derivative test to determine local maxima and minima. $$c = a\left(\frac{-b}{2a}\right)^2 + j \implies j = \frac{4ac - b^2}{4a}$$. If the function f(x) can be derived again (i.e. In general, if $p^2 = q$ then $p = \pm \sqrt q$, so Equation $(2)$ How to find the maximum and minimum of a multivariable function? If the second derivative at x=c is positive, then f(c) is a minimum. The second derivative may be used to determine local extrema of a function under certain conditions. But otherwise derivatives come to the rescue again. The calculus of variations is concerned with the variations in the functional, in which small change in the function leads to the change in the functional value. I have a "Subject: Multivariable Calculus" button. A local maximum point on a function is a point (x, y) on the graph of the function whose y coordinate is larger than all other y coordinates on the graph at points "close to'' (x, y). gives us The solutions of that equation are the critical points of the cubic equation. A point x x is a local maximum or minimum of a function if it is the absolute maximum or minimum value of a function in the interval (x - c, \, x + c) (x c, x+c) for some sufficiently small value c c. Many local extrema may be found when identifying the absolute maximum or minimum of a function. Can you find the maximum or minimum of an equation without calculus? (Don't look at the graph yet!). For the example above, it's fairly easy to visualize the local maximum. $ax^2 + bx + c = at^2 + c - \dfrac{b^2}{4a}$ Examples. On the graph above I showed the slope before and after, but in practice we do the test at the point where the slope is zero: When a function's slope is zero at x, and the second derivative at x is: "Second Derivative: less than 0 is a maximum, greater than 0 is a minimum", Could they be maxima or minima? We say local maximum (or minimum) when there may be higher (or lower) points elsewhere but not nearby. Youre done. DXT. You can do this with the First Derivative Test. x &= -\frac b{2a} \pm \frac{\sqrt{b^2 - 4ac}}{2a} \\ \end{align} The gradient of a multivariable function at a maximum point will be the zero vector, which corresponds to the graph having a flat tangent plane. Any help is greatly appreciated! Math can be tough, but with a little practice, anyone can master it. Direct link to George Winslow's post Don't you have the same n. 1. If the function goes from decreasing to increasing, then that point is a local minimum. Theorem 2 If a function has a local maximum value or a local minimum value at an interior point c of its domain and if f ' exists at c, then f ' (c) = 0. Explanation: To find extreme values of a function f, set f ' (x) = 0 and solve. Now we know $x^2 + bx$ has only a min as $x^2$ is positive and as $|x|$ increases the $x^2$ term "overpowers" the $bx$ term. &= \pm \frac{\sqrt{b^2 - 4ac}}{\lvert 2a \rvert}\\ Step 5.1.2.2. Direct link to Will Simon's post It is inaccurate to say t, Posted 6 months ago. Maximum and Minimum of a Function. original equation as the result of a direct substitution. \end{align}. algebra-precalculus; Share. This figure simply tells you what you already know if youve looked at the graph of f that the function goes up until 2, down from 2 to 0, further down from 0 to 2, and up again from 2 on. "Saying that all the partial derivatives are zero at a point is the same as saying the gradient at that point is the zero vector." The function must also be continuous, but any function that is differentiable is also continuous, so we are covered. So we can't use the derivative method for the absolute value function. is defined for all input values, the above solution set, 0, 2, and 2, is the complete list of critical numbers. f(c) > f(x) > f(d) What is the local minimum of the function as below: f(x) = 2. Take a number line and put down the critical numbers you have found: 0, 2, and 2. 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. y &= a\left(-\frac b{2a} + t\right)^2 + b\left(-\frac b{2a} + t\right) + c 3.) Max and Min of a Cubic Without Calculus. Well think about what happens if we do what you are suggesting. Second Derivative Test. If f(x) is a continuous function on a closed bounded interval [a,b], then f(x) will have a global . You'll find plenty of helpful videos that will show you How to find local min and max using derivatives. 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. These basic properties of the maximum and minimum are summarized . what R should be? \\[.5ex] that the curve $y = ax^2 + bx + c$ is symmetric around a vertical axis. can be used to prove that the curve is symmetric. Solve Now. \end{align}. This video focuses on how to apply the First Derivative Test to find relative (or local) extrema points. By the way, this function does have an absolute minimum value on . There is only one global maximum (and one global minimum) but there can be more than one local maximum or minimum. Trying to understand how to get this basic Fourier Series, Follow Up: struct sockaddr storage initialization by network format-string. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level. A local minimum, the smallest value of the function in the local region. 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. 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. Direct link to bmesszabo's post "Saying that all the part, Posted 3 years ago. quadratic formula from it. This gives you the x-coordinates of the extreme values/ local maxs and mins. as a purely algebraic method can get. . The result is a so-called sign graph for the function.

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This figure simply tells you what you already know if youve looked at the graph of f that the function goes up until 2, down from 2 to 0, further down from 0 to 2, and up again from 2 on.

\r\n

Now, heres the rocket science. we may observe enough appearance of symmetry to suppose that it might be true in general. 5.1 Maxima and Minima. Step 1: Differentiate the given function. Then we find the sign, and then we find the changes in sign by taking the difference again. As in the single-variable case, it is possible for the derivatives to be 0 at a point . 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

\r\n \t
1. \r\n

   Find the first derivative of f using the power rule.

   \r\n
\r\n \t
2. \r\n

   Set the derivative equal to zero and solve for x.

   \r\n\r\n

   x = 0, 2, or 2.

   \r\n

   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

   \r\n\r\n

   is defined for all input values, the above solution set, 0, 2, and 2, is the complete list of critical numbers. Pick a value from each region, plug it into the first derivative, and note whether your result is positive or negative. There is only one equation with two unknown variables. changes from positive to negative (max) or negative to positive (min). The partial derivatives will be 0. You can rearrange this inequality to get the maximum value of $y$ in terms of $a,b,c$. Try it. Plugging this into the equation and doing the If f ( x) < 0 for all x I, then f is decreasing on I . x0 thus must be part of the domain if we are able to evaluate it in the function. Without using calculus is it possible to find provably and exactly the maximum value or the minimum value of a quadratic equation $$ y:=ax^2+bx+c $$ (and also without completing the square)? We try to find a point which has zero gradients . Is the reasoning above actually just an example of "completing the square," 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. We say that the function f(x) has a global maximum at x=x 0 on the interval I, if for all .Similarly, the function f(x) has a global minimum at x=x 0 on the interval I, if for all .. Step 2: Set the derivative equivalent to 0 and solve the equation to determine any critical points. If we take this a little further, we can even derive the standard Thus, to find local maximum and minimum points, we need only consider those points at which both partial derivatives are 0. @return returns the indicies of local maxima. First rearrange the equation into a standard form: Now solving for $x$ in terms of $y$ using the quadratic formula gives: This will have a solution as long as $b^2-4a(c-y) \geq 0$.