Find the local maximum value(s). (Enter your answers as a comma-separated list.) (c) Find the inflection points. smaller x-value (x, y) = larger x-value (x, y) = Find the interval(s) where the function is concave up. (Enter your answer using interval notation.) Find the interval(s) where the function is concave down.Working of a Concavity Calculator. The concavity calculator works on the basis of the second derivative test. The key steps are as follows: The user enters the function and the specific x-value. The calculator evaluates the second derivative of the function at this x-value. If the second derivative is positive, the function is concave up.Question: Given f (x) = (x - 2)^2 (x - 4)^2, determine a. interval where f (x) is increasing or decreasing, b local minima and maxima of f (x) c intervals where f (x) is concave up and concave down, and d. the inflection points of f (x), Sketch the curve, and then use a calculator to compare your answer. If you cannot determine the exact answer ...(W) Consider the function f (x) = a x 3 + b x where a > 0. (a) Consider b > 0. (i) Find the x-intercepts.(ii) Find the intervals on which f is increasing and decreasing. (iii) Identify any local extrema. (iv) Find the intervals on which f is concave up and concave down. (b) Consider b < 0. (i) Find the x-intercepts.(ii) Find the intervals on which f is increasing and decreasing.Hence, what makes \(f\) concave down on the interval is the fact that its derivative, \(f'\), is decreasing. Figure 1.31: At left, a function that is concave up; at right, one that is concave down. We state these most recent observations formally as the definitions of the terms concave up and concave down.Hence, what makes \(f\) concave down on the interval is the fact that its derivative, \(f'\), is decreasing. Figure 1.31: At left, a function that is concave up; at right, one that is concave down. We state these most recent observations formally as the definitions of the terms concave up and concave down.Question: Algabraically determine where each of the following functions are concave up, concave down, increasing and decreasing. Sketch a graph using a graphing calculator and label intervals where increasing/decreasing/ concave up/ concave down. MAKE SURE that graph and calculations agree!! 1) y= (x-2)^3 + 3 ..... x E ALL REAL NUMBERS.Explanation: G(x)= 1/4 x^4-x^3+14 Use the values where the second derivative is zero to set up intervals. Substitute a value into each interval to find where the curve is concave up or down. Concave up on (-∈fty ,0) since f''(x) is positive Concave down on (0,2) since f''(x) is negative Concave up on (2,∈fty ) since f''(x) is positiveA consequence of the concavity test is the following test to identify where we have extrema and inflection points of f. The Second Derivative Test for Extrema is as follows: Suppose that f is a continuous function near c and that c is a critical value of f Then. If f′′ (c)<0, then f has a relative maximum at x=c.Example 3.5.3: Curve sketching. Sketch f(x) = 5 ( x − 2) ( x + 1) x2 + 2x + 4. Solution. We again follow Key Idea 4. We assume that the domain of f is all real numbers and consider restrictions. The only restrictions come when the denominator is 0, but this never occurs. Therefore the domain of f is all real numbers, R.So, the concave up and down calculator finds when the tangent line goes up or down, then we can find inflection point by using these values. Hence, the graph of derivative y = f' (x) increased when the function y = f(x) is concave upward as well as when the derivative y = f' (x) decreased the function is concave downward and the graph ...Free functions and line calculator - analyze and graph line equations and functions step-by-stepThe inflection points of a function are the points where the function changes from either "concave up to concave down" or "concave down to concave up". To find the critical points of a cubic function f(x) = ax 3 + bx 2 + cx + d, we set the second derivative to zero and solve. i.e., f''(x) = 0. 6ax + 2b = 0. 6ax = -2b. x = -b/3aThis 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, ...Luckily, convex and concave are easy to distinguish based on what they look like. A concave function is shaped like a hill or an upside-down U. It's a function where the slope is decreasing. When it's graphed, no line segment that joins 2 points on its graph ever goes above the curve. A convex function, on the other hand, is shaped like a U ...Determine the intervals on which the given function is concave up or down and find the point of inflection. Let f(x)=x(x-8sqrt(x)) The x-coordinate of the point of inflection is The interval on the left of the inflection point is . and on this interval f is Concave Down The interval on the right is . and on this interval f is Concave Up .The Derivative Calculator lets you calculate derivatives of functions online — for free! Our calculator allows you to check your solutions to calculus exercises. It helps you practice by showing you the full working (step by step differentiation). The Derivative Calculator supports computing first, second, …, fifth derivatives as well as ...Here's the best way to solve it. Use a sign chart for F" to determine the intervals on which the function fis concave up or concave down. (Enter your answers using interval notation. If an answer does not exist, enter DNE.) x X-5 concave up X concave down Identify the locations of any inflection points. Then verify your algebraic answers with ...Find the inflection points and intervals of concavity up and down of. f(x) = 3x2 − 9x + 6 f ( x) = 3 x 2 − 9 x + 6. First, the second derivative is just f′′(x) = 6 f ″ ( x) = 6. Solution: Since this is never zero, there are not points of inflection. And the value of f′′ f ″ is always 6 6, so is always > 0 > 0 , so the curve is ...The Derivative Calculator lets you calculate derivatives of functions online — for free! Our calculator allows you to check your solutions to calculus exercises. It helps you practice by showing you the full working (step by step differentiation). The Derivative Calculator supports computing first, second, …, fifth derivatives as well as ...Concavity of graphs of functions - Concave up and down. New Resources. Construct a Conic; Kopie von parabel - parabol; alg2_05_05_01_applet_exp_flvsInflection points are found in a way similar to how we find extremum points. However, instead of looking for points where the derivative changes its sign, we are looking for points where the second derivative changes its sign. Let's find, for example, the inflection points of f ( x) = 1 2 x 4 + x 3 − 6 x 2 . The second derivative of f is f ...We can calculate the second derivative to determine the concavity of the function's curve at any point. Calculate the second derivative. Substitute the value of x. If f " (x) > 0, the graph is concave upward at that value of x. If f " (x) = 0, the graph may have a point of inflection at that value of x. How do you find concave upwards and ...If f ′′(x) < 0 f ′ ′ ( x) < 0 for all x ∈ I x ∈ I, then f f is concave down over I I. We conclude that we can determine the concavity of a function f f by looking at the second derivative of f f. In addition, we observe that a function f f can switch concavity (Figure 6).Find step-by-step Biology solutions and your answer to the following textbook question: Determine where each function is increasing, decreasing, concave up, and concave down. With the help of a graphing calculator, sketch the graph of each function and label the intervals where it is increasing, decreasing, concave up, and concave down. Make sure that your graphs and your calculations agree ...Explore math with our beautiful, free online graphing calculator. Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more.From the source of Khan Academy: Inflection points algebraically, Inflection Points, Concave Up, Concave Down, Points of Inflection. An online inflection point calculator …Subject classifications. A function f (x) is said to be concave on an interval [a,b] if, for any points x_1 and x_2 in [a,b], the function -f (x) is convex on that interval (Gradshteyn and Ryzhik 2000).Given a curve y=f(x), a point of inflection is a point at which the second derivative equals to zero, f''(x)=0, and across which the second derivative changes sign. This means that the curve changes concavity across a point of inflection; either from concave-up to concave-down or concave-down to concave-up. In this section we learn how to find points of …Find step-by-step Biology solutions and your answer to the following textbook question: Determine where each function is increasing, decreasing, concave up, and concave down. With the help of a graphing calculator, sketch the graph of each function and label the intervals where it is increasing, decreasing, concave up, and concave down. Make sure that your graphs and your calculations agree ...Explore math with our beautiful, free online graphing calculator. Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more. Untitled Graph. Save Copy. Log InorSign Up. x − y x + y xy ≥ 0. 1. x 1 y 1 y 2 − 9. 9. − 9. − 7 ...Hence, what makes \(f\) concave down on the interval is the fact that its derivative, \(f'\), is decreasing. Figure 1.31: At left, a function that is concave up; at right, one that is concave down. We state these most recent observations formally as the definitions of the terms concave up and concave down.The major difference between concave and convex lenses lies in the fact that concave lenses are thicker at the edges and convex lenses are thicker in the middle. These distinctions...We say this function \(f\) is concave up. Figure \(\PageIndex{6b}\) shows a function \(f\) that curves downward. As \(x\) increases, the slope of the tangent line decreases. Since the derivative decreases as \(x\) increases, \(f^{\prime}\) is a decreasing function. We say this function \(f\) is concave down.The function is greater than the triangle whose vertex are at (0, 0) ( 0, 0), (2, 0) ( 2, 0) and (1, 1) ( 1, 1). The integral will be greater than the area of this triangle. This trangle has a basis of length 2 2 and a height of 1 1, then an area of 1 1. We could also do it by integral. ∫2 0 f(x)dx ≥∫1 0 xdx +∫2 1 (2 − x)dx = 1 2 + 1 ...Question 296583: find the largest open interval at which function is concave up or concave down and find the location of any points of inflection. f(x)= x^4+8x^3-30x^2+24x-3 Please help with steps Answer by stanbon(75887) (Show Source): You can put this solution on YOUR website!Question: Given f (x)= (x−2)^2 (x−4)^2 , determine a. interval where f (x) is increasing or decreasing, b. local minima and maxima of f (x) c. intervals where f (x) is concave up and concave down, and d. the inflection points of f (x) . Sketch the curve, and then use a calculator to compare your answer. If you cannot determine the exact ...Math. Calculus. Calculus questions and answers. Determine where the given function is concave up and where it is concave down. f (x)=x3+3x2−x−24 Concave up on (−∞,−1), concave down on (−1,∞) Concave down on (−∞,−1) and (1,∞), concave up on (−1,1) Concave up on (−1,∞), concave down on (−∞,−1) Concave down for all x.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 second derivatives of are given by f ' (x) = 2 a x + b f " (x) = 2 a The sign of f " depends on the sign of coefficient a ...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 ...Dec 21, 2020 · Example 5.4.1. Describe the concavity of f(x) = x3 − x. Solution. The first dervative is f ′ (x) = 3x2 − 1 and the second is f ″ (x) = 6x. Since f ″ (0) = 0, there is potentially an inflection point at zero. Since f ″ (x) > 0 when x > 0 and f ″ (x) < 0 when x < 0 the concavity does change from down to up at zero, and the curve is ... For the following function determine: a. intervals where f f f is increasing or decreasing b. local minima and maxima of f f f c. intervals where f f f is concave up and concave down, and d. the inflection points of f f f. f (x) = x 4 − 6 x 3 f(x)=x^{4}-6 x^{3} f (x) = x 4 − 6 x 319. Suppose f (x) is an decreasing, concave down function and you use numeric integration to compute the integral of f over the interval [0, 1]. Put the values of approximations from the least to greatest using n = 50 for Left Endpoint rule L50, Right Endpoint rule R50 and Simpson's rule S5o. a. S50, L50, R50 b. R50, S50, L50 c. L50, S50, R50 d.Determine the intervals on which the given function is concave up or concave down and find the points of inflection. f (x)=2xe−7x (Use symbolic notation and fractions where needed. Give your answer as a comma separated list of points in the form in the form (∗,∗). Enter DNE if there are no points of inflection.) points of ...Use the Concavity Theorem to determine where the given function is concave up and where it is concave down. Also find all inflection points. G (w)=−4w2+16w+15 Concave up for all w; no inflection points Concave down for all w: no inflection points Concavo up on (−2,∞), concave down on (−∞,−2); inflection point (−2,−1) Concavo yp ...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.function-shift-calculator. en. Related Symbolab blog posts. Functions. A function basically relates an input to an output, there's an input, a relationship and an output. For every input... Enter a problem. Cooking Calculators. Cooking Measurement Converter Cooking Ingredient Converter Cake Pan Converter More calculators.function-domain-calculator. concave up. en. Related Symbolab blog posts. Functions. A function basically relates an input to an output, there’s an input, a relationship and an output. For every input... Enter a problem. Cooking Calculators. Cooking Measurement Converter Cooking Ingredient Converter Cake Pan Converter More calculators.0:00 find the interval that f is increasing or decreasing4:56 find the local minimum and local maximum of f7:37 concavities and points of inflectioncalculus ...function-vertex-calculator. en. Related Symbolab blog posts. Functions. A function basically relates an input to an output, there’s an input, a relationship and an output. For every input... Enter a problem. Cooking Calculators. Cooking Measurement Converter Cooking Ingredient Converter Cake Pan Converter More calculators.Subject classifications. A function f (x) is said to be concave on an interval [a,b] if, for any points x_1 and x_2 in [a,b], the function -f (x) is convex on that interval (Gradshteyn and Ryzhik 2000). 1. I have quick question regarding concave up and downn. in the function f(x) = x 4 − x− −−−−√ f ( x) = x 4 − x. the critical point is 83 8 3 as it is the local maximum. taking the second derivative I got x = 16 3 x = 16 3 as the critical point but this is not allowed by the domain so how can I know if I am function concaves up ... The Sign of the Second Derivative Concave Up, Concave Down, Points of Inflection. We have seen previously that the sign of the derivative provides us with information about where a function (and its graph) is increasing, decreasing or stationary.We now look at the "direction of bending" of a graph, i.e. whether the graph is "concave up" or "concave …Excel is a powerful tool that can revolutionize the way you handle calculations. Whether you’re a student, a professional, or just someone who needs to crunch numbers regularly, ma...How do you determine whether the function #f(x) = x^2e^x# is concave up or concave down and its intervals? Calculus Graphing with the Second Derivative Analyzing Concavity of a Function 1 AnswerFinding 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...Inflection points are points where the function changes concavity, i.e. from being "concave up" to being "concave down" or vice versa. They can be found by considering where the second derivative changes signs. In similar to critical points in the first derivative, inflection points will occur when the second derivative is either zero or ...function-domain-calculator. concave up. en. Related Symbolab blog posts. Functions. A function basically relates an input to an output, there’s an input, a ...Explanation: G(x)= 1/4 x^4-x^3+14 Use the values where the second derivative is zero to set up intervals. Substitute a value into each interval to find where the curve is concave up or down. Concave up on (-∈fty ,0) since f''(x) is positive Concave down on (0,2) since f''(x) is negative Concave up on (2,∈fty ) since f''(x) is positiveDetermine the intervals on which the function is concave up or concave down. (Enter your answers using interval notation. Enter EMPTY or o for the empty set.) f (x) = (x - 8) (6 - x) concave up x concave down X Find the points of inflection. (Enter your answers as a comma-separated list.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 …Transcript. Inflection points are points where the function changes concavity, i.e. from being "concave up" to being "concave down" or vice versa. They can be found by considering where the second derivative changes signs. In similar to critical points in the first derivative, inflection points will occur when the second derivative is either ...Determine the intervals on which the given function is concave up or down and find the point of inflection. Let f(x) = x(x−4√x) ... College Algebra Math Help Function Algebra Word Problem Mathematics Ap Calc Ap Calculus Calc Derivatives Calculus 1. RELATED QUESTIONSWolfram Language function: Compute the regions on which an expression is concave up or down. Complete documentation and usage examples. ... Note that at stationary points of the expression, the curve is neither concave up nor concave down. In this case, 0 is a member of neither of the regions: In[5]:= Out[5]=Concave Up Down Calculator. Concave Up Down Calculator - Web if f(x) > 0 for all x on an interval, f'(x) is increasing, and f(x) is concave up over the interval. Web concavity relates to the rate of change of a function's derivative. Our results show that the curve of f ( x) is concaving downward at the interval, ( − 2 3, 2 3).we can therefore determine that: (1) By solving the equation: f '(x) = 0 ⇒ −2xe−x2 = 0. we can see that f (x) has a single critical point for x = 0, this point is a relative maximum since f ''(0) = −2 < 0. Looking at the second derivative, we can see that 2e−x2 is always positive and non null, so that inflection points and concavity ...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-3)(x+2)=0 The critical points are {(x=3),(x=-2 ...Inflection points calculator. An inflection point is a point on the curve where concavity changes from concave up to concave down or vice versa. Let's illustrate the above with an example. Consider the function shown in the figure. From figure it follows that on the interval the graph of the function is convex up (or concave down). On the ...So, since an increasing first derivative indicates concave up, a positive second derivative indicates concave up. Similarly, as a decreasing first derivative indicates concave down, a negative second derivative indicates concave down. The point where the function switches concavity is called the inflection point. Because the function’s first ...of a function can tell you whether the linear approximation will be an overestimate or an underestimate. 1.If f(x) is concave up in some interval around x= c, then L(x) underestimates in this interval. 2.If f(x) is concave down in some interval around x= c, then L(x) overestimates in this interval.Inflection points are found in a way similar to how we find extremum points. However, instead of looking for points where the derivative changes its sign, we are looking for points where the second derivative changes its sign. Let's find, for example, the inflection points of f ( x) = 1 2 x 4 + x 3 − 6 x 2 . The second derivative of f is f ...A concave function can be non-differentiable at some points. At such a point, its graph will have a corner, with different limits of the derivative from the left and right: A concave function can be discontinuous only at an endpoint of the interval of definition.When a function is concave up, the second derivative will be positive and when it is concave down the second derivative will be negative. Inflection points are where a graph switches concavity from up to down or from down to up. Inflection points can only occur if the second derivative is equal to zero at that point. About Andymath.comFind step-by-step Biology solutions and your answer to the following textbook question: Determine where each function is increasing, decreasing, concave up, and concave down. With the help of a graphing calculator, sketch the graph of each function and label the intervals where it is increasing, decreasing, concave up, and concave down. Make sure that your graphs and your calculations agree ...Thus f is concave up from negative infinity to the inflection point at (1, -1), and then concave down from there to infinity. As always, you should check your result on your graphing calculator. Hint: To get a good feel for the look of this function, you need a fairly odd graphing window — try something like xmin = -2, xmax = 4, ymin = -20, ymax = 20.For the function illustrated above, identify the concavity and whether the function is increasing or decreasing on the intervals indicated below. Show transcribed image text. Here's the best way to solve it. Expert-verified.This problem has been solved! You'll get a detailed solution from a subject matter expert that helps you learn core concepts. Question: Find the interval where the function is concave up. Find the. Find the interval where the function is concave up. Find the interval where the function is concave down. Here's the best way to solve it.To determine the intervals where the function f(x) = (x - 14)(1 - x^3) is concave up or concave down and to find the points of inflection, we need to calculate the first and second derivatives of f(x). First, find the first derivative f'(x) by using the product rule: Let u = x - 14 and v = 1 - x^3. Then, u' = 1 and v' = -3x^2.A function f is concave up (or upwards) where the derivative f ′ is increasing. This is equivalent to the derivative of f ′ , which is f ″ , being positive. Similarly, f is concave down (or downwards) where the derivative f ′ is decreasing (or equivalently, f ″ is negative).To determine concavity, analyze the sign of f''(x). f(x) = xe^-x f'(x) = (1)e^-x + x[e^-x(-1)] = e^-x-xe^-x = -e^-x(x-1) So, f''(x) = [-e^-x(-1)] (x-1)+ (-e^-x)(1) = e^-x (x-1)-e^-x = e^-x(x-2) Now, f''(x) = e^-x(x-2) is continuous on its domain, (-oo, oo), so the only way it can change sign is by passing through zero. (The only partition numbers are the zeros of f''(x)) f''(x) = 0 if and only ...Identifying when a function is both concave up and down Understanding change of the second derivative from positive to negative; Practice Exams. Final Exam Math 104: Calculus Status: ...
. Magnificent magnavox
An inflection point is defined as a point on the curve in which the concavity changes. (i.e) sign of the curvature changes. We know that if f " > 0, then the function is concave up and if f " < 0, then the function is concave down. If the function changes from positive to negative, or from negative to positive, at a specific point x = c ...The points where the graph of the function changes from “concave up to concave down” or “concave down to concave up” are called the points of inflection of f (x) . How to calculate point of inflection ? (i) If f ′′(c) exists and f ′′(c) changes sign when passing through x = c , then the pointStep 1. Determine the intervals on which the function is concave up or down. w(t)= tt4−1 +2 (Give your answer as an interval in the form (∗,∗). Use the symbol ∞ for infinity, U for combining intervals, and an appropriate type of parenthesis " (".")", " [","]" depending on whether the interval is open or closed. Enter ∅ if the interval ...Free Pre-Algebra, Algebra, Trigonometry, Calculus, Geometry, Statistics and Chemistry calculators step-by-stepDetermine where each function is increasing, decreasing, concave up, and concave down. With the help of a graphing calculator, sketch the graph of each function and label the intervals where it is increasing, decreasing, concave up, and concave down. Make sure that your graphs and your calculations agree. y = − 2 x 2 + 3 y=\frac{-2}{x^{2}+3 ...Find wher the function is concave up and where it's concave down - identify any inflection points This problem has been solved! You'll get a detailed solution from a subject matter expert that helps you learn core concepts.Determine the intervals on which the function is concave up or down and find the value at which the inflection point occurs. y=11x5−4x4 (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: Incorrect.I'm looking for a concave down increasing-function, see the image in the right lower corner. Basically I need a function f(x) which will rise slower as x is increasing. The x will be in range of [0.10 .. 10], so f(2x) < 2*f(x) is true. Also if. I would also like to have some constants which can change the way/speed the function is concaving.In today’s fast-paced world, efficiency is key. Whether you are a student, professional, or small business owner, finding ways to streamline your tasks can greatly improve producti... The concavity of the graph of a function refers to the curvature of the graph over an interval; this curvature is described as being concave up or concave down. Generally, a concave up curve has a shape resembling "∪" and a concave down curve has a shape resembling "∩" as shown in the figure below. Concave up. Formula to Calculate Inflection Point. We find the inflection by finding the second derivative of the curve's function. The sign of the derivative tells us whether the curve is concave downward or concave upward. Example: Lets take a curve with the following function. y = x³ − 6x² + 12x − 5.Free Pre-Algebra, Algebra, Trigonometry, Calculus, Geometry, Statistics and Chemistry calculators step-by-stepAnswer 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.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.The orientation of a parabola is that it either opens up or opens down; The vertex is the lowest or highest point on the graph; The axis of symmetry is the vertical line that goes through the vertex, dividing the parabola into two equal parts.If \(h\) is the \(x\)-coordinate of the vertex, then the equation for the axis of symmetry is \(x=h\). The maximum or minimum value of a parabola is the ...Apr 24, 2022 · 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. Notice that a function can be concave up regardless of whether it is increasing or decreasing. Details. To visualize the idea of concavity using the first derivative, consider the tangent line at a point. Recall that the slope of the tangent line is precisely the derivative. As you move along an interval, if the slope of the line is increasing, then is increasing and so the function is concave up. Similarly, if the slope of the line is ....
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