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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.

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

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Johns Hopkins University

Missouri State University

Harvey Mudd College

University of Michigan - Ann Arbor

in this problem. You want to look at the relationship between a graph of y equals f of X, and the graph of that functions curvature, which is represented by Y equals cap of X. So we want to know which of A and B depicted on the graph to the left is the graph of why equals f of X and which is the graph of its curvature. So what we're gonna do is we're going to look at the number of inflection or critical points on the graph, and we're gonna compare that with the number of relative Maxima and the relationship between those two things is gonna help us identify witchcraft is which. So let's look at the graph of a first. What we see is we have one point. Actually, one inflection point are one critical point where we change from a positive slope to a negative slope. So we've got one critical point and that one critical point is the relative maximum Here. Now, if we look at B, on the other hand, we have one critical point. Are sorry, one relative, one critical point in one relative maximum. And we also have to inflection points were re changed. Con cavity so total. We have three inflection critical points and one relative maximum. So we need to look and see where our number of inflection or critical points equals the number of relative maximum. See that right here. So what that tells us is that our the graph, corresponding to the number of inflection or critical points, is going to be a graph of f of X. So a represents the graph of y equals F of X and that one, uh, that one inflection point corresponds to the place where we have the maximum curvature. So we'll have a relative maximum for our graph of be, so be represents the curvature. Why equals Kappa of X, See when you're working with these problems, if you've got two graphs, my suggestion is to take a look at the number of inflection, incredible points and the number of there is that a relative maxima and look at the relationship. Your inflection or critical point should equal the, uh, the number of relative maximum on on these two different graph. So, um, the number of critical points or inflection points on your graph of the function should correspond to the number of relative Maxima on the graph of the curvature