Finding Extrema from Graphs

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Question

Finding Extrema from Graphs

In Exercises 11–14, match the table with a graph.

Accepted Answer

Welcome back, everyone. Choose the correct set of values of F of X at the shown values of X in the graph. A says F P equals 0 F Q equals 0, and F R equals -8. For B respectively, we have 0, -8 and 0. For C respectively, we have 80 and 8, and for D respectively, we have 00, and 8. For this problem, let's understand that we're given the graph of the function itself, and we're given three coordinates of X, right? They are labeled as P, Q, and R. We essentially want to evaluate the derivative at each point. We want to estimate f of P, F at Q, and F at R. So let's recall that the derivative is. Simply The slope of the tangent line at the point of tangency. So P is our first point of tangency, and what we want to do is simply draw a tangent line passing through that point, right? What we can notice is that we can draw a horizontal line which is reasonable because we can see that P corresponds to a local maximum point. What we have to understand is that a horizontal line has a zero slope, right? The first derivative at a maximum or a minimum is 0 because the slope of the tangent line is 0. So we can conclude that F at P is equal to 0 because it is a maximum or simply because the slope of the tangent line is 0. Similarly, we can evaluate F at Q. And we can notice that this is a point of minimum, right? Our function was decreasing before Q, after Q it started to increase. So we have a point of a local minimum. And we can notice that we again have a horizontal tangent line, which means that the slope is equal to 0. Finally we went to valid F at r. And in this case, we don't have a horizontal tangent line. In fact, it is a line with a positive slope. If we understand the rise in the run and divide the rise by the run, we can see that our change in Y is positive and the change in X is positive as well. So we have a line with a positive slope. We can only tell that F of R is positive, right? The slope of the tangent line is positive, but we don't know the exact value because we are not given the values. Of the points, right? Relative to the y axis and the X-axis. So now what we're going to do is analyze each option. A says 008, so this is incorrect because F R is positive. So A is not correct, considering B we have 0, -8 and 0. This is also not correct because F Q is 0, it is not ne8 and F R is positive. So there are two errors. For C, we have 80 and 8. This is incorrect. Because we have 00 and being positive, so essentially there's a mistake F of P should be 0. And finally this means that D should be the correct option. It says F P is equal to 0, which is correct. F Q is equal to 0, which is correct, and F R is 8, which is correct. It can be 8 because f of R must be positive. Well then, so the correct answer to this problem corresponds to the answer choice d. Thank you for watching.

Finding Extrema from Graphs

In Exercises 11–14, match the table with a graph.

Key Concepts

Critical Points

First Derivative Test

Extrema

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