2024 Finding concave up and down

For each problem, find the x-coordinates of all points of inflection, find all discontinuities, and find the open intervals where the function is concave up and concave down. 1) y = x3 − 3x2 + 4 x y −8 −6 −4 −2 2 4 6 8 −8 −6 −4 −2 2 4 6 8 Inflection point at: x = 1 No discontinuities exist. Concave up: (1, ∞) Concave down .... Bcrnews

Use concavity and inflection points to explain how the sign of the second derivative affects the shape of a function’s graph. Explain the concavity test for a function over an open …Solution: Since f′(x) = 3x2 − 6x = 3x(x − 2) , our two critical points for f are at x = 0 and x = 2 . We used these critical numbers to find intervals of increase/decrease as well as local extrema on previous slides. Meanwhile, f″ (x) = 6x − 6 , so the only subcritical number is at x = 1 . It's easy to see that f″ is negative for x ...函数的凹凸性可以有多种定义。. 我们这里采取一种比较容易理解的方式来定义。. 1，我们说函数是凹的（concave up），是指函数的切线位于函数的下方。. 从图形上看，函数的切线的斜率是增加的，也就是说 f ′(x) f ′ ( x) 增加。. 由上一节我们知道，函数增加的 ...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.The function is concave down wherever , so we compute and see where it is negative. We have: (a parabola, opening upwards) To find where is negative, we first find its zeros by setting :, so when or , and we conclude that is negative ( is concave down) between them. That is, . The only answer choice completely inside this interval (not outside ...Oct 17, 2019 ... We have the graph of f(x) and need to determine the intervals where it's concave up and concave down as well as find the inflection points.On the interval #(-oo,2)#, we have #f''(x) < 0# so #f# is concave down. On #(2,oo)#, we get #f''(x) >0#, so #f# is concave up. Inflection point. The point #(2, f(2)) = (2,2/e^2)# is the only inflection point for the graph of this function. Find the inflection points and intervals of concavity up and down of f(x) = 2x3 − 12x2 + 4x − 27. Solution: First, the second derivative is f ″ (x) = 12x − 24. Thus, solving 12x − 24 = 0, there is just the one inflection point, 2. Choose auxiliary points to = 0 to the left of the inflection point and t1 = 3 to the right of the ... 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.For $$$ x\gt0 $$$, $$$ f^{\prime\prime}(x)=6x\gt0 $$$ and the curve is concave up. This confirms that $$$ x=0 $$$ is an inflection point where the concavity changes from down to up. Concavity. Concavity describes the shape of the curve of a function and how it bends. The curve can be concave up (convex down), concave down (convex up), or neither.The Second Derivative Test relates to the First Derivative Test in the following way. If , f ″ ( c) > 0, then the graph is concave up at a critical point c and f ′ itself is growing. Since f ′ ( c) = 0 and f ′ is growing at , c, then it must go from negative to positive at . c.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 ...Concavity of Parametric Curves. Recall that when we have a function f, we could determine intervals where f was concave up and concave down by looking at the second derivative of f. The same sort of intuition can be applied to a parametric curve C defined by the equations and . Recall that the first derivative of the curve can be calculated by .Dec 21, 2020 · The second derivative is evaluated at each critical point. When the graph is concave up, the critical point represents a local minimum; when the graph is concave down, the critical point represents a local maximum. The concavity changes at points b and g. At points a and h, the graph is concave up on both sides, so the concavity does not change. At points c and f, the graph is concave down on both sides. At point e, even though the graph looks strange there, the graph is concave down on both sides – the concavity does not change. Find intervals on which the graph of y = x4 - 4x3 - 18x2 + 4 is concave up and intervals on which it If an answer does not exist, enter DNE.) concave up concave down Find the points of inflection. (Order your answers from smallest to largest x, then from smallest to large smaller x-value (x, y) = larger x-value (x, y) = Find any relative maxima ...Step 1: Finding the second derivative. To find the inflection points of f , we need to use f ″ : f ′ ( x) = 5 x 4 + 20 3 x 3 f ″ ( x) = 20 x 3 + 20 x 2 = 20 x 2 ( x + 1) Step 2: Finding all candidates. Similar to critical points, these are points where f ″ ( x) = 0 or where f ″ ( x) is undefined. f ″ is zero at x = 0 and x = − 1 ...To determine whether a function is concave up or concave down using the second derivative, you can follow these steps: Find the second derivative of the function. This involves taking the derivative of the first derivative of the function. The second derivative is often denoted as f''(x) or d²y/dx².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 ...Steps given on how to find Intervals where a Function is Concave up and Concave Down. Directions on how to find inflection points. Multiple of examples of f...Are you looking for a guide to finding an evening dress? Check out our guide to finding an evening dress in this article. Advertisement You may have a pretty good idea of what styl...Question: Question \#5 - Use either the First Derivative or Second Derivative to find which intervals the function is concave up and concave down and all inflection points. (7 points) f (x)=4x4−4x3+5 A) Inflection Pts: B) Intervals Where: Convave Down C) Intervals Where: Concave up. There are 2 steps to solve this one.Anyway here is how to find concavity without calculus. Step 1: Given f (x), find f (a), f (b), f (c), for x= a, b and c, where a < c < b. Where a and b are the points of interest. C is just any convenient point in between them. Step 2: Find the equation of the line that connects the points found for a and b.Can a person choose to be happy? Can you create happiness or do you find it? These 3 steps about how to be happier may help with answers. Finding happiness within yourself can star...Calculus. Find the Concavity f (x)=x^3-6x^2. f(x) = x3 - 6x2. Find the x values where the second derivative is equal to 0. Tap for more steps... x = 2. 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. Interval Notation:Example 1: Determine the concavity of f (x) = x 3 − 6 x 2 −12 x + 2 and identify any points of inflection of f (x). Because f (x) is a polynomial function, its domain is all real numbers. Testing the intervals to the left and right of x = 2 for f″ (x) = 6 x −12, you find that. hence, f is concave downward on (−∞,2) and concave ...Concave up (also called convex) or concave down are descriptions for a graph, or part of a graph: A concave up graph looks roughly like the letter U. A concave down graph is shaped like an upside down U (“⋒”). They tell us something about the shape of a graph, or more specifically, how it bends. That kind of information is useful when it ...For a quadratic function f (x)=ax^2+bx+c, if a>0, then f is concave upward everywhere, if a<0, then f is concave downward everywhere. Wataru · 6 · Sep 21 2014.When it's just you and your kids, how do you find love again, or let love find you as a single parent? Finding love isn’t easy as a single parent, but it’s possible. Learning about...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 ...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.Jul 12, 2015 ... which a function changes concavity, from concave up to concave down, or ... Calculus - Slope, Concavity, Max, Min, and ... Finding the derivative ...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 ...Sep 28, 2022 ... How to determine Concave down and concave up interval and points of inflection and. 2K views · 1 year ago ...more ...Finding Your Way with Clinical Depression All of us feel sad sometimes, but depression is different. Learn how to recognize the signs and symptoms of depression and how to get help...A series of free Calculus Videos and solutions. Concavity Practice Problem 1. Problem: Determine where the given function is increasing and decreasing. Find where its graph is concave up and concave down. Find the relative extrema and inflection points and sketch the graph of the function. f (x)=x^5-5x Concavity Practice Problem 2.Moreover, the point (0, f(0)) will be an absolute minimum as well, since f(x) = x^2/(x^2 + 3) > 0,(AA) x !=0 on (-oo,oo) To determine where the function is concave up and where it's concave down, analyze the behavior of f^('') around the Inflection points, where f^('')=0. f^('') = -(18(x^2-1))/(x^2 + 3)^2=0 This implies that -18(x^2-1) = 0 ...5. Click “Math,” then “Inflection.”. Hit the “diamond” or “second” button, then select F5 to open up “Math.”. In the dropdown menu, select the option that says “Inflection.”. [10] This is—you guessed it—how to tell your calculator to calculate inflection points. 6.Aug 27, 2013 ... How to determine the concavity of functions, and an example involving turtles.Step 1. Given function is f ( x) = x e x. first finding the inflection point. inflection point occur where f ″ ( x) = 0. View the full answer Step 2. Unlock. Answer. Unlock. Previous question Next question.Step 1. Determine the intervals on which the function is concave up or down and find the points of inflection. f (x) = 6x3 – 11x2 + 6 (Give your answer as a comma-separated list of points in the form (* , *). Express numbers in exact form. Use symbolic notation and fractions where needed.) points of inflection: 11 18 Determine the interval on ...The first derivative is f'(x)=3x^2-6x and the second derivative is f''(x)=6x-6=6(x-1). The second derivative is negative when x<1, positive when x>1, and zero when x=1 (and of course changes sign as x increases "through" x=1). That means the graph of f is concave down when x<1, concave up when x>1, and has an inflection point at x=1.example 5 Determine where the cubic polynomial is concave up, concave down and find the inflection points. The second derivative of is To determine where is positive and where it is negative, we will first determine where it is zero. Hence, we will solve the equation for .. We have so .This value breaks the real number line into two intervals, and .The second …For this exercise, decide whether the graph is concave up, concave down, or neither. prealgebra. Perform the transformation shown. Translation 4 units right and 4 units down. earth science. The degradation of landscape by weathering, erosion, and transportation will ultimately reduce the landscape down to _____.Find function concavity intervlas step-by-step. function-concavity-calculator. en. Related Symbolab blog posts. Functions. A function basically relates an input to an output, …Using the results of step 3, find the numbers listed on the number line that lie immediately between an interval that is concave up and one that is concave down. These are the x-values of the ...Determine the intervals on which the function is concave up or down and find the value at which the inflection point occurs. y = 11 x 5 − 4 x 4 (Express intervals in interval notation. Use symbols and fractions where needed.) point of inflection at x = interval on which function is concave up: interval on which function is concave down: IncorrectCalculus questions and answers. Determine the intervals on which the given function is concave up or down and find the point of inflection. Let f (x) = x (x - 5) The x-coordinate of the point of inflection is 225/64 , and on this interval f is The interval on the left of the inflection point is Concave Down The interval on the right is Concave ...Concave downward: $(-\infty, -1)$; Concave upward: $(-1, \infty)$ b. Concave downward: $\left(-\infty, -\sqrt{\dfrac{3}{2}}\right)$ and $\left(1,\sqrt{\dfrac{3}{2}}\right)$; Concave upward: $\left( …Our definition of concave up and concave down is given in terms of when the first derivative is increasing or decreasing. We can apply the results of the previous …This video defines concavity using the simple idea of cave up and cave down, and then moves towards the definition using tangents. You can find part 2 here, ...How can you find a job that you love? Learn 5 tips for finding a job you love at HowStuffWorks. Advertisement Eight hours a day, 40 hours a week, 2,000 hours a year -- for the aver...Walkthrough of Part A. To determine whether f (x) f (x) is concave up or down, we need to find the intervals where f'' (x) f ′′(x) is positive (concave up) or negative (concave down). Let’s first find the first derivative and second derivative using the power rule. f' (x)=3x^2-6x+2 f ′(x) =3x2 −6x+2.Hotwire is one of the go-to sites for online travel searches. But how does Hotwire really work, and are you getting the best travel deal by booking through them? I've gone through ...Calculus. Find the Concavity f (x)=x^4-5x^3. f (x) = x4 − 5x3 f ( x) = x 4 - 5 x 3. Find the x x values where the second derivative is equal to 0 0. Tap for more steps... x = 0, 5 2 x = 0, 5 2. 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 ...Second Derivative and Concavity. Graphically, a function is concave up if its graph is curved with the opening upward (Figure $\PageIndex{1a}$). Similarly, a function is concave down if its graph opens downward (Figure $\PageIndex{1b}$).. Figure $\PageIndex{1}$ This figure shows the concavity of a function at several points. 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 ... The concavity changes at points b and g. At points a and h, the graph is concave up on both sides, so the concavity does not change. At points c and f, the graph is concave down on both sides. At point e, even though the graph looks strange there, the graph is concave down on both sides – the concavity does not change. Homework Statement f(x)=(2x)/((x^2)-25) find concave up and down Homework Equations The Attempt at a Solution I found the second derivative to b -4x((-2x^2)-24)-----((x^2)-25)^2 i found the only inflection point was x=0 (which was correct) I plugged in values on both the right and left side of 0 and determined that f(x) was concave down on all values smaller than 0 with the exception of -5 ...Answer link. First find the derivative: f' (x)=3x^2+6x+5. Next find the second derivative: f'' (x)=6x+6=6 (x+1). The second derivative changes sign from negative to positive as x increases through the value x=1. Therefore the graph of f is concave down when x<1, concave up when x>1, and has an inflection point when x=1.f (x)=3 (x)^ (1/2)e^-x 1.Find the interval on which f is increasing 2.Find the interval on which f is decreasing 3.Find the local maximum value of f 4.Find the inflection point 5.Find the interval on which f is concave up 6.Find the interval on which f is concave down. Anyone can explain? I know the f' (x)=e^-x (3-6x)/2 (x)^ (1/2) calculus. Share.Our definition of concave up and concave down is given in terms of when the first derivative is increasing or decreasing. We can apply the results of the previous section to find intervals on which a graph is concave up or down. That is, we recognize that $\fp$ is increasing when $\fpp>0\text{,}$ etc. Theorem 3.4.4 Test for ConcavityFind the open t-intervals where the parametric Equations are Concave up and Concave DownIf you enjoyed this video please consider liking, sharing, and subscr...Find function concavity intervlas step-by-step. function-concavity-calculator. en. Related Symbolab blog posts. Functions. A function basically relates an input to an ...To graph a function with concave up and down, you can start by finding the concavity using the second derivative test. Then, plot the points where the concavity changes and connect them with a smooth curve. Keep in mind that the function will be increasing when concave up and decreasing when concave down.When asked to find the interval on which the following curve is concave upward $$ y = \int_0^x \frac{1}{94+t+t^2} \ dt $$ What is basically being asked to be done here? Evaluate the integral between $[0,x]$ for some function and then differentiate twice to find the concavity of the resulting function?We say this function f f is concave up. Figure 4.34(b) shows a function f f that curves downward. As x x increases, the slope of the tangent line decreases. Since the derivative decreases as x x increases, f ′ f ′ is a decreasing function. We say this function f f is concave down.Steps given on how to find Intervals where a Function is Concave up and Concave Down. Directions on how to find inflection points. Multiple of examples of f...Sep 28, 2022 ... How to determine Concave down and concave up interval and points of inflection and. 2K views · 1 year ago ...more ...David Guichard (Whitman College) Integrated by Justin Marshall. 4.4: Concavity and Curve Sketching is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts. We know that the sign of the derivative tells us whether a function is increasing or decreasing; for example, when f′ (x)>0, f (x) is …How 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...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 …Theorem 3.4.1Test for Concavity. Let f be twice differentiable on an interval I. The graph of f is concave up if f ′′ > 0 on I, and is concave down if f ′′ < 0 on I. If knowing where a graph is concave up/down is important, it makes sense that the places where the graph changes from one to the other is also important.For each problem, find the x-coordinates of all points of inflection, find all discontinuities, and find the open intervals where the function is concave up and concave down. 1) y = x3 − 3x2 + 4 x y −8 −6 −4 −2 2 4 6 8 −8 −6 −4 −2 2 4 6 8 Inflection point at: x = 1 No discontinuities exist. Concave up: (1, ∞) Concave down ...Oct 17, 2019 ... We have the graph of f(x) and need to determine the intervals where it's concave up and concave down as well as find the inflection points.The graph of a function f is concave down when f ′ is decreasing. That means as one looks at a concave down graph from left to right, the slopes of the tangent lines will be decreasing. Consider Figure 3.4.1 (b), where a concave down graph is shown along with some tangent lines.Apr 12, 2022 · Study the graphs below to visualize examples of concave up vs concave down intervals. It’s important to keep in mind that concavity is separate from the notion of increasing/decreasing/constant intervals. A concave up interval can contain both increasing and/or decreasing intervals. A concave downward interval can contain both increasing and ... The intervals of increasing are x in (-oo,-2)uu(3,+oo) and the interval of decreasing is x in (-2,3). Please see below for the concavities. The function is f(x)=2x^3-3x^2-36x-7 To fd the interval of increasing and decreasing, calculate the first derivative f'(x)=6x^2-6x-36 To find the critical points, let f'(x)=0 6x^2-6x-36=0 =>, x^2-x-6=0 =>, (x … Consequently, to determine the intervals where a function $f$ is concave up and concave down, we look for those values of $x$ where $f''(x)=0$ or $f''(x)$ is undefined. When we have determined these points, we divide the domain of $f$ into smaller intervals and determine the sign of $f''$ over each of these smaller intervals. Concavity of Quadratic Functions. The concavity of functions may be determined using the sign of the second derivative. For a quadratic function f is of the form f (x) = a x 2 + b x + c , with a not equal to 0 The first and …(Enter your answers using interval notation.) f(x) = x + 49 х increasing decreasing Find all relative extrema. (If an answer does not exist, enter DNE.) local minimum at (x, y) = (x, y) = =( local maximum at Find the intervals on which the function is concave up and down. (Enter your answers using interval notation. Anyway here is how to find concavity without calculus. Step 1: Given f (x), find f (a), f (b), f (c), for x= a, b and c, where a < c < b. Where a and b are the points of interest. C is just any convenient point in between them. Step 2: Find the equation of the line that connects the points found for a and b. 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 … The turning point at ( 0, 0) is known as a point of inflection. This is characterized by the concavity changing from concave down to concave up (as in function ℎ) or concave up to concave down. Now that we have the definitions, let us look at how we would determine the nature of a critical point and therefore its concavity. Hence the function f f f is concave-up for x > 1 x>1 x > 1 and concave-down for x < 1 x<1 x < 1. x = 1 x=1 x = 1 is point of inflection of the function f f f. These results can be seen from the graph of the function f f f in Figure 2 2 2. Figure 2. Concave up and down. \small\text{Figure $2$. Concave up and down.} Figure 2. Concave up and down.Sep 13, 2020 · Finding the Intervals where a Function is Concave Up or Down f(x) = (x^2 + 3)/(x^2 - 1)If you enjoyed this video please consider liking, sharing, and subscri... if 0 < x < e^(-15/56) then f is concave down; if x > e^(-15/56) then f is concave up; x=e^(-15/56) is a (falling) inflection point To analyze concavity and inflection points of a twice differentiable function f, we can study the positivity of the second derivative. In fact, if x_0 is a point in the domain of f, then: if f''(x_0)>0, then f is concave up in a …Polynomial graphing calculator. This calculator graphs polynomial functions. All polynomial characteristics, including polynomial roots (x-intercepts), sign, local maxima and minima, growing and decreasing intervals, points of inflection, and concave up-and-down intervals, can be calculated and graphed.When f'(x) is zero, it indicates a possible local max or min (use the first derivative test to find the critical points) When f''(x) is positive, f(x) is concave up When f''(x) is negative, f(x) is …However, as we decrease the concavity needs to switch to concave up at $x \approx - 0.707$ and then switch back to concave down at $x = 0$ with a final switch to concave up at $x \approx 0.707$. Once we hit $x = 1$ the graph starts to increase and is still concave up and both of these behaviors continue for the rest of the graph.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-verified. Use the Concavity Theorem to determine where the given function is concave up and where it is concave down. Also find all inflection points. q(x)= 3x3+2x+8 Concave down for all x; no inflection points Concave up for all k; no inflection points Concave up on (−∞,0), concave down on (0,∞); inflection point (0,8) Concave up .... White river electric coop

Dec 29, 2020 · The graph of the parametric functions is concave up when $\frac{d^2y}{dx^2} > 0$ and concave down when $\frac{d^2y}{dx^2} <0$. We determine the intervals when the second derivative is greater/less than 0 by first finding when it is 0 or undefined. Once the second parametric derivative is found, any value of t can be plugged into the second derivative in order to determine the concavity of the curve at that specific value of t. In Calculus 1 you learn that a function is concave up when the second derivative is positive, and the function is concave down when the second derivative is ...Consequently, to determine the intervals where a function $f$ is concave up and concave down, we look for those values of $x$ where $f''(x)=0$ or $f''(x)$ is undefined. When we have determined these points, we divide the domain of $f$ into smaller intervals and determine the sign of $f''$ over each of these smaller intervals.The Second Derivative Test relates to the First Derivative Test in the following way. If , f ″ ( c) > 0, then the graph is concave up at a critical point c and f ′ itself is growing. Since f ′ ( c) = 0 and f ′ is growing at , c, then it must go from negative to positive at . c.A function that increases can be concave up or down or both, if it has an inflection point. The increase can be assessed with the first derivative, which has to be > 0. The …Are you looking for a guide to finding an evening dress? Check out our guide to finding an evening dress in this article. Advertisement You may have a pretty good idea of what styl...Apr 24, 2022 ... Graphically, a function is concave up if its graph is curved with the opening upward (Figure 2.7.1a). Similarly, a function is concave down if ... It can easily be seen that whenever f'' is negative (its graph is below the x-axis), the graph of f is concave down and whenever f'' is positive (its graph is above the x-axis) the graph of f is concave up. Point (0,0) is a point of inflection where the concavity changes from up to down as x increases (from left to right) and point (1,0) is ... Finding the Intervals where a Function is Concave Up or Down f(x) = (x^2 + 3)/(x^2 - 1)If you enjoyed this video please consider liking, sharing, and subscri...When f'(x) is zero, it indicates a possible local max or min (use the first derivative test to find the critical points) When f''(x) is positive, f(x) is concave up When f''(x) is negative, f(x) is concave down When f''(x) is zero, that indicates a possible inflection point (use 2nd derivative test)Apr 24, 2022 ... Graphically, a function is concave up if its graph is curved with the opening upward (Figure 2.7.1a). Similarly, a function is concave down if ....

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