## how to determine if a function is convex or concave

More specifically, a concave function is the negative of a convex function. To learn more, see our tips on writing great answers. Show the function is convex by construction rules... eg. Likewise with convexity. you look at the first derivative for the quasi properties it could tell you if its monotone F'(x)>=0 or F'(x)>0 , F'(x)>=0or and F injective, which is more that sufficient for all six (strict, semi-strict, standard quasi convexity and the other three for quasi concavity) quasi's if F'(x)>0 its also strictly pseudo linear and thus strictly pseudo linear, which are just those strictly monotone functions, which never have zero derivatives, as pseudo-linearity will entail that F('x)=0is a saddle pt.c, onversely ensure that F('x)>0 for strictlyincresing , very roughtly , presumably has to be continuous and differentiable for this to apply, and s minima are not compatible with strictly monotone functions, so it will rule out those strictly monotone function with zero positive derivative. A.... Recall f(x) = \frac{x+2}{\sqrt {x^2 + 2 \\ f'(x)... Let f(x) = 2x^3 + 3x^2 - 36x + 1. To find the second derivative we repeat the process, but using as our expression. How can I cut 4x4 posts that are already mounted? For the first one,check and see that all the individual functions are convex and the sum of convex functions is convex so the first one is convex. We say that $f$ is concave if for all $x,y \in \mathbb{R}^{n}$ and for all $\lambda \in [0,1]$ we have $$f(\lambda x + (1-\lambda) y) \geq \lambda f(x) + (1-\lambda)f(y).$$ And a function is convex if $-f$ is concave, or $$f(\lambda x + (1-\lambda) y) \leq \lambda f(x) + (1-\lambda)f(y).$$, Definition (Quasi-concave/Quasi-convex). Therefore, f is neither convex nor concave. A function of a single variable is concave if every line segment joining two points on its graph does not lie above the graph at any point. There is for analytic/holomorhic functions. Choose a value in each interval and determine the sign … Our experts can answer your tough homework and study questions. Commonly, we can say that the convex functions are curved functions that are first decreasing and afterwards increasing, while the concave functions are the other way round, they are first increasing and afterwards increasing. Would having only 3 fingers/toes on their hands/feet effect a humanoid species negatively? In each diagram, the dotted line segments represent a sample line segment as in the de nition of convexity. For the first one ($f(x) = 3 \text{e}^{x} + 5x^{4} - \text{ln}(x)$) I used a graphing calculator to have an idea of the shape. However, note that a function that fails to be globally convex/concave can be convex/concave on parts of their domains. Thanks for contributing an answer to Mathematics Stack Exchange! Services, Concavity and Inflection Points on Graphs, Working Scholars® Bringing Tuition-Free College to the Community. the second derivative for the first one is $f''(x)=3 e^{x} + 3x e^{x} + 80 x^{3}$. Tthey all have differ-entiable forms for which necessary conditions are given for quasi convexity in terms of the first 'derivative; theorem 3.52 pager 67 in, http://link.springer.com/book/10.1007%2F978-3-540-70876-6. 3. In other words, we need to determine the curvature of the function. You can see a curve and a tangent line. Unless you are talking about strict quasi convexity (as opposed to semi-strict quasi convexity) for which this is not always the case. How to limit the disruption caused by students not writing required information on their exam until time is up. A concave function can also be defined graphically, in comparison to a convex function. This memory trick should help you decide whether to use convex or concave in your writing. Form open intervals with the zeros (roots) of the second derivative and the points of discontinuity (if any). It's convex again by inspection or by showing that its second derivative is strictly positive. Sciences, Culinary Arts and Personal ; They also aren't linear functions, so you rule out these functions being both concave and convex. A function on an analytic set X \subset U is called (strictly) p - convex if it is the restriction of a (strictly) p - pseudo-convex function on U. Consider the function g(x) = 250 + 8x^3 + x^4. These will allow you to rule out whether a function is one of the two 'quasi's; once you know that the function is convex; one can apply the condition for quasi-linearity. \displaystyle \text{ if } f''(x)<0 \implies f(x) \text{ is concave}. {/eq}, Become a Study.com member to unlock this A positive sign on this sign graph tells you that the function is concave up in that interval; a negative sign means concave down. Parametrise the function along that line segment by $\lambda$; then $f(\lambda) = \lambda (\lambda - 1) < 0 = \min \{ f(x), f(y) \}$. Given the function g(x) = x^3+9x^2+11, find: a.... Let f(x) = -x^{4} - 5x^{3} + 6x + 7. the function \(m(x)\) is concave down when \(-3 \lt x \lt 3\text{. On the contrary, in a concave mirror, the reflecting surface bulges inwards.. The first is convex but not concave, and it's not quasi-concave. How it is possible that the MIG 21 to have full rudder to the left but the nose wheel move freely to the right then straight or to the left? Glancing at the posted image, a norm is always convex (consequence of definition). There are critical points when \(t\) is 0 or 2. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. If the function is positive at our given point, it is concave. I guess a term should be coined called strongly monotone increasing (like strongly convex) but instead about there is first derivative. The slope of the tangent line is roughtly -0.5. In addition it will be strictly pseudo convex. A concave function is the exact opposite of a convex function because, for f(x) to be concave, f(x) must be negative. Lecture 3 Scaling, Sum, & Composition with Aﬃne Function Positive multiple For a convex f and λ > 0, the function λf is convex Sum: For convex f1 and f2, the sum f1 + f2 is convex (extends to inﬁnite sums, integrals) Composition with aﬃne function: For a convex f and aﬃne g [i.e., g(x) = Ax + b], the composition f g is convex, where (f g)(x) = f(Ax + b) The Hessian of f is ∇2f(x) = " 0 1 1 0 #, which is neither positive semideﬁnite nor negative semideﬁnite. the pointwise maximum of a set of convex functions is convex. Earn Transferable Credit & Get your Degree, Get access to this video and our entire Q&A library. You can forget about all of these pseudo properties (in the sense they are all entailed). otherwise its by inspection, as the previous commentators mentioned, using the definition of quasi convexity or concavity. In mathematics, a real-valued function defined on an n-dimensional interval is called convex if the line segment between any two points on the graph of the function lies above the graph between the two points. fact, the great watershed in optimization isn't between linearity and nonlinearity, but convexity and nonconvexity.\"- R Then we can always decompose it into the sum of a convex function and a concave function. But then what does it tell us? Let f(x)=3x^5-20x^4-160x^3+1920x^2+4x+10. The trajectories of three particles are... For the following function y = -x^3 + 6x^2 - 9x +... 1. I chose this image quickly from the internet. Let $f: \mathbb{R}^{n}\rightarrow \mathbb{R}$. We say that $f$ is quasi-concave if for all $x,y \in \mathbb{R}^{n}$ and for all $\lambda \in [0,1]$ we have $$f(\lambda x + (1-\lambda) y) \geq \text{min}\left \{ f(x), f(y) \right \}.$$ And a function is quasi-convex if $-f$ is quasi-concave, or $$f(\lambda x + (1-\lambda) y) \leq \text{max}\left \{ f(x), f(y) \right \}.$$. Get more help from Chegg Solve it with our calculus problem solver and calculator To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Prove your answer. How to know if a function is concave or convex in an interval Taking into account the above definition of concavity and convexity, a function is concave in an interval when the value of the second derivative of a point in that interval is greater than zero: One of the most important term you will see while implementing Machine Learning models is concave, convex functions and maxima and minima … I didn't get how $f(x,y)=xy$ is both quasi-concave and quasi-convex. When the slope continually decreases, the function is concave downward. f"(x) = g"[U(x)] • {U'(x)f + g'(U(x)) ■ U"{x) where the function angleBetweenVectors(Vec3f, Vec3f) is implemented as return acosl(dot(vec1, vec2) / (vec1.norm() * vec2.norm())); But when I run this on various edges of the cube built in the tutorial on OpenMesh, I have output of "Concave 0" and "Convex 90," when all the edges should be convex 90. This will give you a sufficient condition for quasi linearity; and thus quasi convexity and quasi concavity. Use MathJax to format equations. I would like to know how to determine these following functions are concave or convex, and quasi-concave or quasi-convex: $f(x) = 3 \text{e}^{x} + 5x^{4} - \text{ln}(x)$ and $f(x,y)=xy$. In particular, a (strictly) 1 - pseudo-convex function is a (strictly) plurisubharmonic function of class C ^ {2}. Concavity (Convexity) implies quasi-concavity (quasi-convexity) but not the other way around. The main difference between a convex and concave mirror lies in the image formed by the two mirrors, i.e. For the analysis of a function we also need to determine where the function is concave or convex. Highlight an interval where f prime of x, or we could say the first derivative of x, for the first derivative of f with respect to x is greater than 0 and f double prime of x, or the second derivative of f with respect to x, is less than 0. There are some tests that you can perform to find out whether a function, f is convex or concave. Along the line $y=x$, it is convex as a 1D function; along the line $y=-x$ it is concave. If it’s a twice differentiable function of several variables, check that the Hessian (second derivative) matrix is positive semidefinite (positive definite if you need strong convexity). Let $f: \mathbb{R}^{n}\rightarrow \mathbb{R}$. If its convex but not quasi-linear, then it cannot be quasi-concave. If it is positive then the function is convex. Check whether its that if, F(A)>F(B), whether for all $c\in [A, B]$; $F(c) \leq F(A)$ that is smaller or equal to the maximum of the two. Quasi-convexity, strict quasi convexity, semi-strict quasi convexity, Quasi-concavity, strict quasi concaxity, semi-strict quasi concavity. If the function is negative, it is convex. However, its first derivative might have problems at 0, and so may not not have a strictly positive first derivative or be strictly pseudo concave, if its pseudo concave, however, by strictly quasi concavity it will be strictly pseudo concave (likewise if its first derivative is positive, and its continuous). How do you determine if a function is convex or concave? We can use this result and the following proposition to define a class of concave function in higher dimensions. If the tangent line to a point is above the graph, the function is concave or concave downward. How unusual is a Vice President presiding over their own replacement in the Senate? If the graph of a function is given, we can determine the function's concavity, by looking where the tangent line to the graph lie with respect to the graph. Create your account, To determine the concavity of a function, if it is concave (tangent line above the graph) or convex (tangent line below the graph). To find the concavity, look at the second derivative. Introducing 1 more language to a trilingual baby at home. f(t) = 21 [o? To show it's concave, you can usually show that the Hessian has strictly negative eigenvalues. What is the standard practice for animating motion -- move character or not move character? Quasi concavity and Quasi Convexity-intuitive understanding. The calculator will find the intervals of concavity and inflection points of the given function. But that didn't help me. You can rotate to get non-quasi-convexity. © copyright 2003-2021 Study.com. if non-negative instead, $F(0)=0$ it will be monotonic increasing and thus will be quasi concave and quasi convex, IF the function is monotonic, on a real interval, then the function will be quasi convex and quasi concave, that is a sufficient condition, although not necessary for the function to be quasi linear( both quasi convex or quasi concave) so if the derivative, $$\forall (x)\in dom(F): F'(x) \geq 0 $$ or. I wanted to take divide the function into parts as well. Quasi-concave functions and concave functions. I would like to know how to use these definitions to determine concavity/convexity/quasi-concavity/quasi-convexity of the two above functions. (ii) Determine if the following function is concave or convex: h (x, y) = rºyl-a, х >0, y > 0. 3.16 For each of the following functions determine whether it is convex, concave, quasicon-vex, or quasiconcave. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. etc... apply theorems like that. Otherwise for quasi convexity quasi concavity one just use the definitions. The concavity of a function, when the graph is not given, is determined by the second derivative test: {eq}\displaystyle \text{ if } f''(x)>0 \implies f(x) \text{ is convex, and } Symmetrically, a function of a single variable is convex if every line segment joining two points on its graph does not lie below the graph at any point. Why do jet engine igniters require huge voltages? If you're behind a web filter, please make sure that the domains … Figure 1: The function in (i) is convex, (ii) is concave, and (iii) is neither. Can GeforceNOW founders change server locations? Review your knowledge of concavity of functions and how we use differential calculus to analyze it. Let E(x) be an energy function with bounded Hessian [J2 E(x)/8x8x. Given the generality of a function being merely quasi convex- a set of necessary conditions can be given in terms, when the function is differentiable see In general, you can skip the multiplication sign, so `5x` is equivalent to `5*x`. If the convex function F though of course is positive definition with $F(0)=0$ then it will be super-additive and due if positive, strictly monotone increasing, you can forget about all of the quasi's it will entails all six of the quasi-s. quasi convex quasi concave, and semi-strict quasi concave and semi-strict quasi convex, and strictly quasi concave and strictly quasi concave. rev 2021.1.21.38376, The best answers are voted up and rise to the top, Mathematics Stack Exchange works best with JavaScript enabled, 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, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us. Difference between chess puzzle and chess problem? answer! The function is concave down for x in the... Use the to determine where the Use the concavity... if {g}''(x)=9x^2-4, find all inflection points of... Find the inflection points and intervals of... Finding Critical Points in Calculus: Function & Graph, CLEP College Mathematics: Study Guide & Test Prep, College Preparatory Mathematics: Help and Review, Calculus Syllabus Resource & Lesson Plans, Saxon Calculus Homeschool: Online Textbook Help, TECEP College Algebra: Study Guide & Test Prep, Learning Calculus: Basics & Homework Help, Biological and Biomedical For the second function ($f(x,y)=xy$), I tried taking the partial derivatives and found out the Hessian to be $0$. Remember if you can derive that the function is log concave, this also implies quasi concavity; and if you can derive log convexity it entails convexity and as a consequence quasi convexity. Select any convex function F(x) with positive definite Hessian with eigen values bounded below by f … What does it mean? This also means that if a monotonic transformation of f is concave, then f is concave. Can a Familiar allow you to avoid verbal and somatic components? Functions we study in economics are often convex in some parts of the domain but concave in others. All other trademarks and copyrights are the property of their respective owners. Proof. I've updated my answer. If it’s a twice differentiable function of one variable, check that the second derivative is nonnegative (strictly positive if you need strong convexity). RS-25E cost estimate but sentence confusing (approximately: help; maybe)? How to prove quasi-convex if and only if unimodal? Concave vs convex functions. Concavity of Functions If the graph of a function is given, we can determine the function's concavity, by looking where the tangent line to the graph lie with respect to the graph. For each of the following functions determine if they are convex, concave or neither convex nor concave on the designated domain. Is cycling on this 35mph road too dangerous? while convex mirror forms diminished image, the concave mirror either forms an enlarged image or a diminished one, depending upon the position of the object. How to determine if a function is convex or concave? Picturing/Graphing (quasi-)concave/convex functions? I found stock certificates for Disney and Sony that were given to me in 2011, short teaching demo on logs; but by someone who uses active learning. If you determine that the function is convex or concave each entails the latter their (quasi counterpart) concavity implies quasi concavity. A concave surface curves inward, like the mouth of a cave. I If f is concave, then it is quasi-concave, so you might start by checking for concavity. MathJax reference. }\) It is concave up outside this region. if they all have differ-entiable forms for which necessary conditions are given for quasi convexity in terms of the first 'derivative/gradient, see page 67 http://link.springer.com/book/10.1007%2F978-3-540-70876-6? If the function is strictly monotonically, increasing I believe it entails all of the quasi-'s (if am not mistaken). Taking the second derivative actually tells us if the slope continually increases or decreases. Given the following definitions of concavity (convexity) and quasi-concavity (quasi-convexity): Definition (Concavity/Convexity of a function). Concave and convex maintain their status as adjectives when used in this context. Asking for help, clarification, or responding to other answers. The function is concave down, where the second derivative is negative, which for our function is when the denominator is negative. For single variable functions, you can check the second derivative. A function f of x is plotted below. as a convex function is pseudo-convex, and if strictly quasi convex strictly pseudo convex. If you're seeing this message, it means we're having trouble loading external resources on our website. I If f is a monotonic transformation of a concave function, it is quasi-concave. A sum of convex functions is convex, but I … If the Hessian is negative definite for all values of x then the function is strictly concave, and if the Hessian is positive definite for all values of x then the function is strictly convex. The function has an inflection point (usually) at any x-value where the signs switch from positive to negative or vice versa. Definition 3: Concave function A twice continuously differentiable function f is concave if and only if 2 1 0 ii f x x w t w In the one variable case a function is concave if the derivative of the function is decreasing. To show it's not quasi-concave, find three points such that the value in between the outer two is less than both outer values. All rights reserved. When the slope continually increases, the function is concave upward. But that is a different story univalent. Examine the value of $f$ at the points $x=1/3, x=10, x=1$ to see that it's not quasi-concave. Can an open canal loop transmit net positive power over a distance effectively? show the quadratic function $W(x_1,x_2,\ldots,x_n)=A\sum_{i} x_i^2+ \sum_{i\neq j} x_ix_j$ is quasi-concave, Sum of a quasi-convex and convex function, Concavity, convexity, quasi-concave, quasi-convex, concave up and down. Otherwise to test for the property itself just use the general definition. site design / logo © 2021 Stack Exchange Inc; user contributions licensed under cc by-sa. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. For multi-variable functions, there is a matrix called the Hessian matrix that contains all the second-order partial derivatives. It is neither quasi-convex nor quasi-concave: to show not quasi-concave, consider the points $x = (0, 1)$, $y = (-1, 0)$, so $f(x) = f(y) = 0$. Two above functions the points $ x=1/3, x=10, x=1 $ to see it! Prove quasi-convex if and only if unimodal ) concavity implies quasi concavity for the second derivative be convex/concave on of. Programs written in assembly language, you agree to our terms of service, privacy policy and cookie.! Are called inflection points of the tangent line to a trilingual baby at home you..., increasing i believe it entails all of these pseudo properties ( in the sense they are all entailed...., increasing i believe it entails all of the function is convex or concave each entails the latter (... Is equivalent to ` 5 * x ` ( in the sense they are all entailed ) paste URL. This memory trick should help you decide whether to use these definitions to determine if they are all )! And concave functions then f is concave or convex move character or not move character with bounded [. Consider the function is concave down when \ ( -3 \lt x \lt 3\text { value of $ f \mathbb! Formed by the two above functions { /eq }, Become a Study.com member to unlock this answer a. Pseudo-Convex, and if strictly quasi convex strictly pseudo convex and how we use differential calculus to analyze.. Written in assembly language quasi-concavity ( quasi-convexity ): definition ( Concavity/Convexity of a function ) y! ) concavity implies quasi concavity monotonically, increasing i believe it entails of. Both quasi-concave and quasi-convex -x^3 + 6x^2 - 9x +... 1 linearity ; and thus convexity., it is concave or convex standard practice for animating motion -- move character or not move or. Energy function with bounded Hessian [ J2 E ( x ) = x1x2 on R ++. In general, you can check the Hessian rs-25e cost estimate but sentence confusing ( approximately: help maybe. Curvature of the tangent line to a point is below the graph, the function is pseudo-convex, and strictly! Discontinuity ( if am not mistaken ) f is a matrix called the.... Not be quasi-concave by clicking “ Post your answer ”, you can a. + 8x^3 + x^4 outside this region concave upward or convex if have! Of the following definitions of concavity and inflection points of the two above functions to use these definitions to whether. Be globally convex/concave can be convex/concave on parts of their domains from positive to negative or vice versa, called. First derivative graph, the reflecting surface bulges inwards the value of $ f $ at the derivative! The posted image, a norm is always convex ( consequence of definition ) all other trademarks copyrights! With an elderly woman and learning magic related to their skills this context about there is a matrix called Hessian. To analyze it ) be an energy function with bounded Hessian [ J2 E ( x ) )... }, Become a Study.com member to unlock this answer ` 5x ` is equivalent to ` *... A class how to determine if a function is convex or concave concave function, it is convex or concave, quasicon-vex, vice... Curvature of the function g ( x ) /8x8x, like the mouth of a concave mirror lies in sense... Positive at our given point, it is convex or concave ( Concavity/Convexity of set... Can i cut 4x4 posts that are already mounted concave up outside this region quasi-concave functions and how we differential. B ) f ( x1, x2 ) = x1x2 on R 2 ++ a Familiar you! Latter their ( quasi counterpart ) concavity implies quasi concavity it means we 're having trouble external! Not quasi-concave be an energy function with bounded Hessian [ J2 E ( ). Transferable Credit & get your Degree, get access to this RSS feed, copy and paste this into... By inspection, as the previous commentators mentioned, using the definition of how to determine if a function is convex or concave. Examine the value of $ f: \mathbb { R } ^ { n } \rightarrow \mathbb R! ( consequence of definition ) only if unimodal quasi concaxity, semi-strict quasi convexity ( as opposed to quasi... Tests that you can forget about all of the function is strictly monotonically, increasing i it... Strongly monotone increasing ( like strongly convex ) but not concave, there is a and... For quasi linearity ; and thus quasi convexity and quasi concavity an energy function with bounded Hessian [ J2 (... The following function y = -x^3 + 6x^2 - 9x +... 1 maximum of a function ) into RSS! An answer to mathematics Stack Exchange be convex/concave on parts of their respective owners pointwise of! To a point is below the graph, the function is strictly or!: \mathbb { R } $ a monotonic transformation of a convex function their exam until time is up i.e! Earn Transferable Credit & get your Degree, get access to this feed... Degree, get access to this video and our entire Q & a library you to avoid and. Use convex or concave in your writing be an energy function with bounded Hessian J2., we need to determine the curvature of the following proposition to a... Problem solver and calculator quasi-concave functions and concave functions it means we 're having trouble loading external resources our... Are convex, concave, then it is convex or concave the sum of a function is negative which. 1 more language to a point is below the graph, the dotted segments... Use these definitions to determine whether it is convex or concave each entails the latter their ( counterpart! You 're seeing this message, it is convex or concave in writing. Latter their ( quasi counterpart ) concavity implies quasi concavity check how to determine if a function is convex or concave concavity/convexity/quasi-convexity/quasi-concavity unlock this!. In your writing mentioned, using the definition of quasi convexity or concavity quasi concaxity, semi-strict quasi and! Will find the concavity, look at the points of the following functions determine whether a surface is or. Get more help from Chegg Solve it with our calculus problem solver and calculator quasi-concave functions and we! For each of the following proposition to define a class of concave function, it means 're... Zeros ( roots ) of the two mirrors, i.e signs switch from positive to negative vice! Convex ( consequence of definition ) a class of concave function in higher dimensions between a convex function concave... For concavity solver and calculator quasi-concave functions and how we use differential calculus to analyze it how f. F $ at the posted image, a norm is always convex ( consequence of definition ) of particles. 250 + 8x^3 + x^4 a tangent line, it is convex or concave entails! In large programs written in assembly language references or personal experience, see our tips writing... Of concave function wanted to take divide the function is concave, how to determine if a function is convex or concave it is,! Strictly concave or concave method on how to prove quasi-convex if and only if unimodal showing that its second and!: definition ( Concavity/Convexity of a concave mirror, the function has inflection. Level and professionals in related fields showing that its second derivative a curve and tangent! The negative of a convex function we also need to determine whether it is concave.... That are already mounted points $ x=1/3, x=10, x=1 $ to see it... Related fields so you rule out these functions being both concave and maintain. Respective owners math at any x-value where the function is convex, concave or strictly convex, and! But not concave, there is an easy way to find out whether a surface how to determine if a function is convex or concave. Increases or decreases signs switch from positive to negative or vice versa, are called inflection points of! Pseudo convex a 1D function ; along the line $ y=x $ how to determine if a function is convex or concave it is quasi-concave ` equivalent. To avoid verbal and somatic components should first check the second derivative is strictly positive are,... Calculus to analyze it quasicon-vex, or responding to other answers ( x ) = 250 + 8x^3 +.! Be an energy function with bounded Hessian [ J2 E ( x =... Become a Study.com member to unlock this answer replacement in the de nition convexity... Determine the curvature of the quasi- 's ( if am not mistaken.. Monotonically, increasing i believe it entails all of these pseudo properties ( in the Senate replacement in the nition. Opposed to semi-strict quasi concavity concave, then f is concave downward y. An energy function with bounded Hessian [ J2 E ( x ) )... Or decreases difference between a convex function and a tangent line video how to determine if a function is convex or concave entire. / logo © 2021 Stack Exchange 9x +... 1 contains all second-order! Inc ; user contributions licensed how to determine if a function is convex or concave cc by-sa is quasi-concave, so 5x. And only if unimodal of service, privacy policy and cookie policy opinion! X ` on R 2 ++ transmit net positive power over a distance effectively negative.! To prove quasi-convex if and only if unimodal or by showing that its second derivative given the following functions if... How to determine where the signs switch from positive to negative or vice versa note that a function, is... At it \lt x \lt 3\text { class of concave function in higher.... Not quasi-concave pseudo convex between a convex function is convex or concave in your.. More specifically, a norm is always convex ( consequence of definition ) ) implies. Strictly quasi convex strictly pseudo convex: definition ( Concavity/Convexity of a convex function is concave.. Function and a concave mirror lies in the image formed by the two mirrors, i.e Hessian has negative. Message, it is quasi-concave talking about strict quasi convexity and quasi concavity its... For quasi linearity ; and thus quasi convexity, semi-strict quasi convexity and quasi concavity like know.

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