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Two graphs, $a$ and $b,$ are shown. One is a curve $y=f(x)$ and the other is the graph of its curvature function $y=\kappa(x) .$ Identify each curve and explain your choices.

$b$ is a graph of $y=f(x)$ and $a$ is a graph of its curvature $y=\kappa(x)$.

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Missouri State University

Harvey Mudd College

University of Nottingham

Boston College

in this problem we're going to do is look at the difference in relationship between the graph of a function. Why equals F of X and the graph of its curvature. Why equals cap of X? So we know that in the picture on the left ah, that the grass and be correspond to one of these two. Why equals f of X or y equals cap of X? And we want to do is figure out which of Andy represents the function y equals f of X and which represents the functions curvature, cap of x. So the way that I like to approach this type of problem is by considering the number of inflection or critical points of the graph has we'll talk about why in a minute and considering the number of relative Maxima, the graph has. So the relationship between these two things ah, the inflection or critical points represents the places where our graphics changing direction. So remember that the curvature cap of X corresponds to how quickly the graph is changing direction. So if we look at the inflection and critical points where the graph is either changing from a positive to a negative slope or where we have a change in khan cavity that represents a place where our graphics changing directions and that should correspond to a relative Maxima. And our graph of capital affects. Because any place that the graph is is changing direction at an inflection point, that should correspond to a high, ah, larger value for curvature. So let's consider our graphs of A and B. So if we look at a, we see that we have one to three for five critical points and then to inflection points were changing from Khan gave up to cock a down or vice versa. So total we have 1234567 inflection or critical points for the graph of a. And if we look at the number of relative Maxima, we have one to three. And if we look at the graph of B, we see that we have one critical point and we have to Inflection points were changing con cavity. They would be right about here. First we go from concave up to con cave down and then get down to con cave up. So total we have three inflection or critical points and just one relative maximum there in the middle. And what we want to dio is consider where there is a relationship between the number of relative Maxima and the number of inflection or critical points. So if we have three inflection or critical points in our graph of B, that is going to correspond to three relative Maxima in the graph of the curvature function, so we can use this information to state. That be is the graph of the function. Why equals f of X and A is the graph of its curvature. Why equals capital of X And this is because of the relationship we see between the number of inflection or critical points and the number of relative Maxima.