John riley minor corrections 25 july 2016 ucla econ. Find the second derivative and calculate its roots. The rst function is said to be concave up and the second to be concave down. Of particular interest are points at which the concavity changes from up to down or down to. Concave and convex functions 1concaveandconvexfunctions 1.
Critical point c is where f c 0 tangent line is horizontal, or f c undefined tangent line is vertical f x indicates if the function is concave up or down on certain intervals. If d0fx 0y x 0 0 for all y, then x 0 is a global max of f. Concave andquasiconcave functions 1 concaveandconvexfunctions 1. Maxima for differentiable concave functions from elementary calculus we know that f. At the breakpoints, the function isnt differentiablehas no unique tangent line, but wed still want to call the function concaveup. A function f is concave over a convex set if and only if the function. To study the concavity and convexity, perform the following steps.
Monotonicity theorem let f be continuous on the interval. 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. The preceding examples are special cases of power functions, which have the general form y x p, for any real value of p, for x 0. Similarly, a function is concave down if its graph opens downward figure 1b. In particular, if p 1, then the graph is concave up, such as the parabola y x2. Is f concave first note that the domain of f is a convex set, so the definition of concavity can apply the functions g and f are illustrated in the following figures. This means the graph of f has no jumps, breaks, or holes in it. A justification should make use of a known calculus test. You can locate a functions concavity where a function is concave up or down and inflection points where the concavity switches from positive to negative or vice versa in a few simple steps. Calculus conditions for concave functions of a single variable. However, we want to find out when the slope is increasing or decreasing, so we either need to look at the formula for the slope the first derivative and decide, or we need to use the second derivative. The axes for g are shown in perspective, like those for f, to make.
The problem with this is that a monotonic transformation of a concave or convex function need not be concave or convex. Geometrically speaking, a function is concave up if its graph lies above its tangent lines. So our task is to find where a curve goes from concave upward to concave downward or vice versa. There is more than one right way to sketch the graph. The acceleration of a moving object is the derivative of its velocity that is, the second derivative of its. If on an interval, then is concave up on that interval. Determine where the given function is increasing and decreasing. The function is therefore concave at that point, indicating it is a local. Concavity introduction using derivatives to analyze. Graphically, a function is concave up if its graph is curved with the opening upward a in the figure. Find where its graph is concave up and concave down. This is the common way to define cu and cd in operations research, a subfield of applied math that includes a. R is concave convex if and only if its restriction to every line segment of.
Increasing and decreasing functions characterizing functions behaviour typeset by foiltex 2. Find the relative extrema and inflection points and sketch the graph of the function. Demonstrating the 4 ways that concavity interacts with increasingdecreasing, along with the relationships with the first and second derivatives. The sign of the second derivative \fx\ tells us whether \f\ is increasing or decreasing. Find all xvalues of the relative extrema of fix and as relative maximum, minimum, or neither. You can locate a function s concavity where a function is concave up or down and inflection points where the concavity switches from positive to negative or vice versa in a few simple steps. In other words, you can draw the graph of f without lifting your pen or pencil.
If p 0, then the graph starts at the origin and continues to rise to infinity. If f is continuous ata and f changes concavity ata, the point. Roughly speaking concavity of a function means that the graph is above chord. If a function f has a derivative that is in turn differentiable, then its second derivative is the derivative of the derivative of f, written as fif fa exists, we say that f is twice differentiable.
The following method shows you how to find the intervals of concavity and the inflection points of. When the slope continually increases, the function is concave upward. When the slope continually decreases, the function is concave downward. Consider the functions of each question and apply knowledge of the second derivative to. Most of these questions require you to justify your answers. Increasing and decreasing functions, min and max, concavity studying properties of the function using derivatives typeset by foiltex 1. Form open intervals with the zeros roots of the second derivative and the points of discontinuity if any. Let f be the function defined by fx xelx for all real numbers x. While they are both increasing, their concavity distinguishes them. Notice that a function can be concave up regardless of whether it is increasing or decreasing. More lessons for calculus math worksheets a series of free calculus videos and solutions. A function is said to be concave down on an interval if the graph of the function is below the tangent at each point of the interval.
Recall that a realvalued function f is concave if and only if its domain is a convex set a. John riley minor corrections 25 july 2016 concave functions in economics 1. A function is concave up when its gradient increases as its values increase. 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. With concave functions, solving maximization problems is so much easier. How to locate intervals of concavity and inflection points. Concavity problems with formulas, solutions, videos. When the second derivative is positive, the function is concave upward. Study the intervals of concavity and convexity of the following function. Use this quiz to determine the concavity of functions. 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 b in the figure. Taking the second derivative actually tells us if the slope continually increases or decreases. Fromnow on we will assume thatx is aconvex subset of rn. Lets try and untangle what these terms mean by drawing some pictures. For each problem, find the xcoordinates of all points of inflection and find the open intervals where the function is concave up and concave down. Concave and convex functions1 washington university. The second derivative tells us if the slope increases or decreases. Lesson 2convexity and concavity of functions of one and. By proposition 2 it follows that the upper contour sets of f. Therefore d jjlnx is concave and so, by proposition 1, fx is concave.
If you can find a vector satisfying the first order conditions for a maximum, then you have. R is concave convex if and only if its restriction to every line segment of rn is concave convex function of one variable. Increasing and decreasing functions, min and max, concavity. This figure shows the concavity of a function at several points. Concave up concave down in case of the two functions above, their concavity relates to the rate of the increase. Functions and their graphs input x output y if a quantity y always depends on another quantity x in such a way that every value of x corresponds to one and only one value of y, then we say that y is a function of x, written y f x. For each problem, find the xcoordinates of all points of inflection, find all discontinuities, and find the open intervals where the function is concave up and concave down. Using the derivative to analyze functions f x indicates if the function is. There are two types of concavity that are particularly useful in calculus.
At the static point l 1, the second derivative l o 0 is negative. We know that the sign of the derivative tells us whether a function is increasing or decreasing. Remember, we can use the first derivative to find the slope of a function. Tests for local extrema and concavity in all of these problems, each function f is continuous on its domain. A function is concave down if its graph lies below its tangent lines. In each part, sketch the graph of the function f with the stated properties, and discuss the signs of f0and f00.