2024 Concave upward and downward calculator

An inflection point exists at a given x -value only if there is a tangent line to the function at that number. This is the case wherever the first derivative exists or where there’s a vertical tangent. Plug these three x- values into f to obtain the function values of the three inflection points. The square root of two equals about 1.4, so .... Hart's funeral home tahlequah

The calculator will try to find the domain, range, x-intercepts, y-intercepts, derivative, integral, asymptotes, intervals of increase and decrease, critical (stationary) points, extrema (minimum and maximum, local, relative, absolute, and global) points, intervals of concavity, inflection points, limit, Taylor polynomial, and graph of the single-variable function.Calculus. Calculus questions and answers. Discuss the concavity of the graph of the function by determining the open intervals on which the graph is concave upward or downward. Seo E (x) = -3x3 - 6x2 + 8 Interval --00X CX00 Sign of '' (x) 0 FO Conclusion Concave upward Concave downward.Calculus. Find the Concavity f (x)=x^4-8x^2+8. f(x) = x4 - 8x2 + 8. Find the x values where the second derivative is equal to 0. Tap for more steps... x = 2√3 3, - 2√3 3. The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.Isoquant Curve: The isoquant curve is a graph, used in the study of microeconomics , that charts all inputs that produce a specified level of output. This graph is used as a metric for the ...If the graph of f(x) is concave upward or concave downward at a point where the graph has a horizontal tangent line, then there is a local minimum or local maximum, respectively, at that point. Lesson 11.2 described the relationship between a second derivative and a function.Solution. We see that the function is not constant on any interval. The function is increasing where it slants upward as we move to the right and decreasing where it slants downward as we move to the right. The function appears to be increasing from \displaystyle t=1 t = 1 to \displaystyle t=3 t = 3 and from \displaystyle t=4 t = 4 on.A Concave function is also called a Concave downward graph. Intuitively, the Concavity of the function means the direction in which the function opens, concavity describes the state or the quality of a Concave function. For example, if the function opens upwards it is called concave up and if it opens downwards it is called concave down.Concave Upward Or Downward Calculator . Determining the type of concavity of a parametric curve. Substitute any number from the interval...The line is at y = tf (a) + (1t)f (b) And (for concave upward) the line should not be below the curve: For concave downward the line should not be above the curve ( becomes ): And those are the actual definitions of concave upward and concave downward. Derivatives can help! The derivative of a function gives the slope.Share a link to this widget: More. Embed this widget » This problem has been solved! You'll get a detailed solution from a subject matter expert that helps you learn core concepts. Question: Determine where the function is concave upward and where it is concave downward. (Enter your answer using interval notation. If an answer does not exist, enter DNE.) 9 (x) - 1 + x2 concave upward concave downward.Exercise 1: Find the intervals where the function in the given graph is concave upward or concave downward, and any points of inflection. Concave up: Concave down: Point of inflection: Exercise 2: Find the intervals where the given function is concave upward or concave downward, and any points of inflection. f(x) = x4 - 4x3 + 10How to identify the x-values where a function is concave up or concave downPlease visit the following website for an organized layout of all my calculus vide...Find the interval(s) where the following function is concave down. Graph to double check your answer.The function y = f (x) is called convex downward (or concave upward) if for any two points x1 and x2 in [a, b], the following inequality holds: If this inequality is strict for any x1, x2 ∈ [a, b], such that x1 ≠ x2, then the function f (x) is called strictly convex downward on the interval [a, b]. Similarly, we define a concave function.Question: Discuss the concavity of the graph of the function by determining the open intervals on which the graph is concave upward or downward. f(x) = -2x3 - 7x2 + 1 = Interval - < X < < X < 00 Sign of f'(x) f" f" 0 Conclusion Concave upward Concave downward J6 Points] DETAILS PREVIOUS ANSWERS LARCAAPCALC2 8.6.019. Discuss the concavity of the graph of theThe curve can be concave up (convex down), concave down (convex up), or neither. In mathematical terms, a function $$$ f(x) $$$ is concave up on an interval if the second derivative $$$ f^{\prime\prime}(x) $$$ is positive at each point of the interval and concave down if it is negative at each point of the interval.There is an easy way to tell whether the graph of a quadratic function opens upward or downward: if the leading coefficient is greater than zero, the parabola opens upward, and if the leading coefficient is less than zero, the parabola opens downward. Study the graphs below: Figure %: On the left, y = x 2. On the right, y = - x 2.Figure 4.5.2: The function f has four critical points: a, b, c ,and d. The function f has local maxima at a and d, and a local minimum at b. The function f does not have a local extremum at c. The sign of f ′ changes at all local extrema. Using Figure 4.5.2, we summarize the main results regarding local extrema.1. Suppose you pour water into a cylinder of such cross section, ConcaveUp trickles water down the trough and holds water in the tub. ConcaveDown trickles water away and spills out, water falling down. In the first case slope is <0 to start with, increases to 0 and next becomes > 0. In the second case slope is >0 at start, decreases to 0 and ...Functions. A function basically relates an input to an output, there's an input, a relationship and an output. For every input... Read More. Save to Notebook! Sign in. Free Functions End Behavior calculator - find function end behavior step-by-step.Polynomial graphing calculator. This page helps you explore polynomials with degrees up to 4. The roots (x-intercepts), signs, local maxima and minima, increasing and decreasing intervals, points of inflection, and concave up-and-down intervals can all be calculated and graphed.The First Derivative Test. Corollary 3 of the Mean Value Theorem showed that if the derivative of a function is positive over an interval I then the function is increasing over I. On the other hand, if the derivative of the function is negative over an interval I, then the function is decreasing over I as shown in the following figure. Figure 1.Graphically, a function is concave up if its graph is curved with the opening upward (Figure 1a). Similarly, a function is concave down if its graph opens downward (Figure 1b). Figure 1. This figure shows the concavity of a function at several points. Notice that a function can be concave up regardless of whether it is increasing or decreasing.Final answer. Consider the following graph. Step 1 of 2: Determine the intervals on which the function is concave upward and concave downward. Enable Zoom/Pan 7.5 -10 10 -7.5 -15 Answer 2 Points Keypad Keyboard Shortcuts Separate multiple entries with a comma. Selecting a radio button will replace the entered answer value (s) with the radio ...١٥‏/٠٤‏/٢٠٢٢ ... Find predesigned Concave Up Down Calculator Ppt Powerpoint Presentation Ideas Design Inspiration Cpb PowerPoint templates slides, graphics, ...A function is said to be concave on an interval if, for any points and in , the function is convex on that interval (Gradshteyn and Ryzhik 2000).calculus. Determine the open intervals on which the graph of the function is concave upward or concave downward. f ( x) = x + 8 x − 7. f (x)=\frac {x+8} {x-7} f (x) = x−7x+8. . physics. In a galaxy far, far away, a planet composed of an incompressible liquid of uniform mass density. ρ.Calculus. Find the Concavity f (x)=x^3-12x+3. f (x) = x3 − 12x + 3 f ( x) = x 3 - 12 x + 3. Find the x x values where the second derivative is equal to 0 0. Tap for more steps... x = 0 x = 0. The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the ...Expert Answer. 100% (1 rating) Transcribed image text: Find the open intervals where the function is concave upward or concave downward. Find any inflection points. Select the correct choice below and fill in any answer boxes within your choice. O A. The function is concave up on and concave down on (Type your answer in interval notation.Consider the following graph. Step 1 of 2: Determine the intervals on which the function is concave upward and concave downward. Enable Zoom/Pan TO A 10 7.5 Keyboard Shortcu Separate multiple entries with a comma. Selecting a radio button will replace the entered answer value(s) with the radio button value.So, by determining where the function is concave up and concave down, we could quickly identify a local maximum and two local minimums. Nice! In this video lesson, we will learn how to determine the intervals of concavity (concave upward and downward), locate inflection points, and use the second derivative test to identify relative extrema. ...f is concave up. b) If, at every point a in I, the graph of y f x always lies below the tangent line at a, we say that-f is concave down. (See ﬁgure 3.1). Proposition 3.4 a) If f is always positive in the interval I, then f is concave up in that interval. b) If f is always negative in the interval I, then f is concave down in that interval.The first and the second derivative of a function can be used to obtain a lot of information about the behavior of that function. For example, the first derivative tells us where a function increases or decreases and where it has maximum or minimum points; the second derivative tells us where a function is concave up or down and where it has inflection points.Compute dy dt. dy dt = t − 1. Use the following equation taken from the reference: dy dx = dy dt dx dt. Substitute our computations: dy dx = t −1 t +1. Use the following equation taken from the reference: d2y dx2 = d( dy dx) dt dx dt. To compute d(dy dx) dt, we use the quotient rule:1. When asked to find the interval on which the following curve is concave upward. y =∫x 0 1 94 + t +t2 dt y = ∫ 0 x 1 94 + t + t 2 d t. What is basically being asked to be done here? Evaluate the integral between [0, x] [ 0, x] for some function and then differentiate twice to find the concavity of the resulting function? calculus.Concave downward, downward, is an interval, or you're gonna be concave downward over an interval when your slope is decreasing. So g prime of x is decreasing or we can …Concave up The following curves are examples of curves which are concave up; that is they bend up or open upwards like a cup. The tangents to the curve sit underneath the curve. ... y concave down 0 concave up The graph of y = x3 +x. 0 2 x -2 y -1 12 3 Mathematics Learning Centre, University of Sydney 3 Point of inﬂection that is a ...This problem has been solved! You'll get a detailed solution from a subject matter expert that helps you learn core concepts. Question: Determine where the function is concave upward and where it is concave downward. (Enter your answer using interval notation. If an answer does not exist, enter DNE.) 9 (x) - 1 + x2 concave upward concave downward.Calculus. Calculus questions and answers. 1.) a Determine where the function is concave upward and where it is concave downward. (Enter your answer using interval notation. If an answer does not exist, enter DNE.) g (x) = -x2 + 8x + 2 concave upward concave downward b Determine where the function is concave upward and where it is concave downward.Final answer. Consider the following graph. 1) Determine the intervals on which the function is concave upward and concave downward. 2) Determine the x -coordinates of any inflection point (s) in the graph. None of these. Concave up: (−∞,−6)∪ (−1,3); Concave down: (−6,−1)∪(3,∞) x -value (s) of inflection point (s): x = −6,x ...A function is said to be concave on an interval if, for any points and in , the function is convex on that interval (Gradshteyn and Ryzhik 2000). See also Convex Function Explore with Wolfram|Alpha. More things to try: …Question: Find the intervals on which the graph of f is concave upward, the intervals on which the graph of f is concave downward, and the inflection points. f (x)= ln (x^2-8x+41) Find the intervals on which the graph of f is concave upward, the intervals on which the graph of f is concave downward, and the inflection points. f (x)= ln (x^2-8x+41)label local min and max, points of inflection, and clearly indicate intervals of concave up and down and increase and decrease. Graphs: If the graph is concave ...A: We know that volume of a rectangular prism is the product of area of the base and the height of the…. The vertical shift of the function is y = -cos (÷ + -4 is 4 units downward. O True False. Determine the intervals on which the graph of y = f (x) is concave up or concave down, and find the…. Let f (x)=2x2+8x+7.Sep 9, 2015 · Using the second derivative test, f(x) is concave up when x<-1/2 and concave down when x> -1/2. Concavity has to do with the second derivative of a function. A function is concave up for the intervals where d^2/dx^2f(x)>0. A function is concave down for the intervals where d^2/dx^2f(x)<0. First, let's solve for the second derivative of the ... This problem has been solved! You'll get a detailed solution from a subject matter expert that helps you learn core concepts. Question: Determine where the graph of the given function is concave upward and concave downward. Find the coordinates of all inflection points. f (x) = x3 + 6x2 + x + 9.If we are trying to understand the shape of the graph of a function, knowing where it is concave up and concave down helps us to get a more accurate picture. Of …A curve is concave up if it has the shape of a bowl that would hold water. It is concave down if it has the shape of an upside down bowl. This is illustrated below. y= f(x) concave up y= (x) concave down The graph of a function can be concave up on some intervals and concave down on others. The graph shown below is concave down on the intervals ...Recognizing the different ways that it can look for a function to paass through two points: linear, concave up, and concave down.Substitute any number from the interval (0,∞) into the second derivative and evaluate to determine the concavity. Tap for more steps... Concave up on (0,∞) since f ''(x) is positive. The graph is concave down when the second derivative is negative and concave up when the second derivative is positive. Concave down on (−∞,0) since f ''(x ...So g, so concave upward means that your first derivative increasing, increasing, which means, which means that your second derivative is greater than zero. And concave downward is the opposite. Concave downward, downward, is an interval, or you're gonna be concave downward over an interval when your slope is decreasing.a)Determine where the graph of the function is concave upward and where it is concave downward. Also, find all inflection points of the function b) Determine where the graph of the function is concave upward and where it is concave downward. Also, find all inflection points of the function. c) Given f ( x) = x 2 - 4 x , Find the intervals on ...View more at www.MathAndScience.com.In this lesson, you will learn what factors determine if a parabola (quadratic equation) opens up or down in the xy plane...Step 1 of 2: Determine the intervals on which the function is concave upward and concave downward. 151 7.5 x On 10 10 -7.5 -15) Get more help from Chegg Solve it with our Calculus problem solver and calculator.Final answer. Determine the open intervals on which the graph of the function is concave upward or concave downward. (Enter your answers using interval notation. If an answer does not exist, enter DNE.) y = -x3 + 3x2 - 8 concave upward concave downward Determine the open intervals on which the graph of the function is concave upward or …In order to find what concavity it is changing from and to, you plug in numbers on either side of the inflection point. if the result is negative, the graph is concave down and if it is positive the graph is concave up. Plugging in 2 and 3 into the second derivative equation, we find that the graph is concave up from and concave down from .Calculus questions and answers. Find the intervals on which the graph of f is concave upward, the intervals on which the graph of f is concave downward, and the infle f (x) =-x4 + 16x3-16x + 5 For what interval (s) of x is the graph of f concave upward? Select the correct choice below and, if necessary, fill in the answer box to choice.Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might haveJan 22, 2016. For a quadratic function ax2 +bx + c, we can determine the concavity by finding the second derivative. f (x) = ax2 + bx +c. f '(x) = 2ax +b. f ''(x) = 2a. In any function, if the second derivative is positive, the function is concave up. If the second derivative is negative, the function is concave down.2. [2/2 points) PREVIOUS ANSWERS ASK YOUR TEACHER DETAILS MY NOTES Determine the open intervals on which the graph is concave upward or concave downward. (Enter your answers using interval notation. If an answer does not exist, enter DNE.) concave upward (-00,0) "( 3,00) concave downwardFinal answer. Determine the open intervals on which the graph of the function is concave upward or concave downward. (Enter your answers using interval notation. If an answer does not exist, enter DNE.) y = -x3 + 3x2 - 8 concave upward concave downward Determine the open intervals on which the graph of the function is concave upward or …Question: In Exercises 5 through 12, determine where the graph of the given function is concave upward and concave downward. Find the coordinates of all inflection points. ... Solve it with our Pre-calculus problem solver and calculator. Not the exact question you're looking for? Post any question and get expert help quickly. Start learning ...Recall that the first derivative of the curve C can be calculated by dy dx = dy/dt dx/dt. If we take the second derivative of C, then we can now calculate intervals where C is concave up or concave down. (1) d2y dx2 = d dx(dy dx) = d dt(dy dx) dx dt. Now let's look at some examples of calculating the second derivative of parametric curves. Free Functions Concavity Calculator - find function concavity intervlas step-by-stepYes it would, assuming that the function is defined at the point. An inflection point only requires: 1) that the concavity changes and. 2) that the function is defined at the point. You can think of potential inflection points as critical points for the first derivative — i.e. they may occur if f" (x) = 0 OR if f" (x) is undefined.concave down if $f$ is differentiable over an interval $I$ and $f'$ is decreasing over $I$, then $f$ is concave down over $I$ concave up if $f$ is differentiable over an interval $I$ and $f'$ is increasing over $I$, then $f$ is concave up over $I$ concavity the upward or downward curve of the graph of a function ...The second derivative tells whether the curve is concave up or concave down at that point. If the second derivative is positive at a point, the graph is bending upwards at that point. Similarly, if the second derivative is negative, the graph is concave down. This is of particular interest at a critical point where the tangent line is flat and ... Calculus Examples. Step-by-Step Examples. Calculus. Applications of Differentiation. Find the Concavity. f (x) = −x2 + 2x + 6 f ( x) = - x 2 + 2 x + 6. Find the x x values where the second derivative is equal to 0 0. Tap for more steps... No solution.Concavity introduction Google Classroom About Transcript Sal introduces the concept of concavity, what it means for a graph to be "concave up" or "concave down," and how this relates to the second derivative of a function. Created by Sal Khan. Questions Tips & Thanks Want to join the conversation? Sort by: Top Voted 13NixonF 10 years agoExpert Answer. You are given the graph of a function f Determine the intervals where the graph of fis concave upward and where it is concave downward. (Enter your answers using interval n concave upward concave downward Find all inflection points of f, if any. (If an answer does not exist, enter DNE.) (x, y)Calculus. Find the Concavity f (x)=x^3-12x+3. f(x) = x3 - 12x + 3. Find the x values where the second derivative is equal to 0. Tap for more steps... x = 0. The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.See full list on calculator-online.net Calculus questions and answers. You are given the graph of a function f. (i) Determine the intervals where the graph of f is concave upward and where it is concave downward. (Enter your answers using interval notation.) concave upward concave downward Find the inflection point of fs if any. (If an answer does not exist, enter DNE.) (x,y)= (.The derivative is: y' = 3x 2 − 12x + 12. The second derivative is: y'' = 6x − 12. And 6x − 12 is negative up to x = 2, positive from there onwards. So: f (x) is concave downward up to x = 2. f (x) is concave upward from x = 2 on. And the inflection point is at x = 2: Calculus Index. y. f x. ′= is shown above. (a) Use the graph of f ′ to determine whether the graph of f is concave up, concave down, or neither on the interval.value is positive, the function is concave upward in that interval; negative, the function is concave downward in the interval. Definition of a Point of Inflection: If a graph of a continuous function has a tangent line at a point where the concavity changes from upward to downward (or downward to upward), then that point is a point of inflection.A: according to graph given function is concave upward for x>0 and concave downward for x<0 Q: Determine the open intervals on which the graph of the function is concave upward or concave… A: y=x+2sin x Let take first derivative y=x+2cscxy'=1-2cotxcscxNow take second…Finding where a curve is concave up or down. You guessed it, it isn't enough to know what concave up or concave down curves look like! We need to be able to find where curves are concave up or down. A curve can have some parts that are concave up and other parts that are concave down, and it's useful to be able to work out which is which, even ... Expert Answer. 100% (1 rating) Transcribed image text: Identify the open intervals on which the graph of the function is concave upward or concave downward. Assume that the graph extends past what is shown. 10- 1 00 8- 6- 4 2 2 4 6 6 8 10 -10._-8-6-4 -2 0 -2- ܠܐ 4 6 1 -8 10- Note: Use the letter for union. To enter , type infinity.٢٠‏/١٢‏/٢٠٢٠ ... Figure 3.4.4: A graph of a function with its inflection points marked. The intervals where concave up/down are also ...Select the correct choice below and, if necessary, fill in the answer box (es) to complete your choice. O A. The function is concave upward on the interval (s) The function is never concave downward. (Type your answer in interval notation. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed.) OB.

A curve is concave up if it has the shape of a bowl that would hold water. It is concave down if it has the shape of an upside down bowl. This is illustrated below. y= f(x) concave up y= (x) concave down The graph of a function can be concave up on some intervals and concave down on others. The graph shown below is concave down on the intervals .... Gun shows in south carolina

Consequently, to determine the intervals where a function $f$ is concave up and concave down, we look for those values of $x$ where $f^{\prime\prime}(x) = 0$ or $f^{\prime\prime}(x)$ is undefined. When we have determined these points, we divide the domain of $f$ into smaller intervals and determine the sign of \(f^{\prime\prime ...Polynomial graphing calculator. This page helps you explore polynomials with degrees up to 4. The roots (x-intercepts), signs, local maxima and minima, increasing and decreasing intervals, points of inflection, and concave up-and-down intervals can all be calculated and graphed.Question Video: Determining the Type of Concavity of a Parametric Curve Mathematics. Question Video: Determining the Type of Concavity of a Parametric Curve. Consider the parameric curve 𝑥 = 1 + sec 𝜃 and 𝑦 = 1 + tan 𝜃. Determine whether this curve is concave up, down, or neither at 𝜃 = 𝜋/6.Expert Answer. 1. Concave upward => (-5,1)U (4,infinity) . Concav …. Step 1 of 2: Determine the intervals on which the function is concave upward and concave downward. Step 2 of 2: Determine the x-coordinates of any inflection point (s) in the graph.However, how do we know that if our estimation is an overestimate or an underestimate? We calculate the second derivative and look at the concavity. Concave up vs Concave down. If the second derivative of the function is greater than 0 for values near a, then the function is concave up. This means that our approximation will be an underestimate.1. Below is a chart that gives some information regarding a twice-differentiable function fx. (The "n/a" in the chart means "not applicable.") *<-4 x= -4 -4<x<0 x=0 0<x< 4 x = 4 4 <x n/a -3 n/a 1 n/a 5 n/a negative 0 positive 0 negative 0 positive Concavity n/a n/a n/a Fill in the last row of the chart (the four empty spaces) with the proper concavity (either "concave-up" or "concave-down ...What Is the Concavity Function? The concavity of a function is the convex shape formed when the curve of a function bends. There are two types of concavities in a graph i.e. concave up and concave down. How To Calculate the Inflection Point. The calculator determines the inflection point of the given point by following the steps mentioned below:Explain, in two different ways, without using the rules of differentiation, why the derivative of the constant function f (x)=2 f (x) = 2 must be f^ {\prime} (x)=0 f ′(x)= 0. [Hint: Think of the slope of the graph of a constant function, and also of the instantaneous rate of change of a function that stays constant.] GoodSportsBuys.com is an ...Find the open intervals where the function is concave upward or concave downward. Find any inflection points Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice A. The function is concave up on and concave down on (Type your answers in interval notation. Use a comma to separate answers as needed) B.Calculus questions and answers. Find the intervals on which the graph of f is concave upward, the intervals on which the graph of f is concave downward, and the infection points. f (x) = -x^4 + 8x^3 - 8x + 7 For what interval (s) of x is the graph of f concave upward? Select the correct choice below and, if necessary, fill in the answer box to ...In particular, f x x (0, 0) = 2 > 0 ‍ , and the fact that this is positive means f (x, y) ‍ looks like it has upward concavity as we travel in the x ‍ -direction. On the other hand, the second partial derivative with respect to y ‍ is a negative constant:Concave down: If a function is concave up (like a parabola), what is 𝑓 ñ is doing. If 𝑓 is concave up, then 𝑓 ñ is increasing. If 𝑓 is concave down, then 𝑓 ñ is decreasing. This leads us to the following… 𝑓 ñ ñ P0 means 𝑓 is concave up. 𝑓 ñ ñ O0 means 𝑓 is concave down. 1. Find the intervals of concavity for ...Determine the open intervals on which the graph is concave upward or concave downward. (Enter your answers using interval notation. If an answer does not exist, enter DNE.) g(x)=18x^2-x^3There are two types of concavity: concave upward and concave downward. If the second derivative of a function f is increasing, {eq}f''(x)>0 {/eq}, then it is called concave upward.When negative, it's concave down. The point where this changes is the point of inflection. The point of inflection is equal to when the second derivative is equal to zero. Let's work with the function for a bit to determine the second derivative: f (x) = 3x2 − x3 3. f '(x) = 2 ⋅ 3x − 3 x2 3. f '(x) = 6x − x2.Transcribed image text: Consider the following graph. Step 1 of 2: Determine the intervals on which the function is concave upward and concave downward. Enable Zoom/Pan 5 7.5 10 10 -7.5 -151.Get the free "Inflection Points" widget for your website, blog, Wordpress, Blogger, or iGoogle. Find more Mathematics widgets in Wolfram|Alpha.A curve is concave up if it is a curve that dips down and up again. It will look like a valley. This is the part of the roller coaster where you go really fast down to the bottom and then you go ...You'll get a detailed solution from a subject matter expert that helps you learn core concepts. Question: Suppose f (x)=x3−4x2−5x Find intervals on which the function is concave upward and intervals on which it is concave downward. a) Concave upward on (-∞, -0.9246) ∪ (0, ∞) ; concave downward on (-0.9246, 0) b) Concave upward on (0 ....

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