Each of Exercises 89–92 shows the graphs of the first and seco...

Question

Each of Exercises 89–92 shows the graphs of the first and second derivatives of a function y=f(x). Copy the picture and add to it a sketch of the approximate graph of f, given that the graph passes through the point P.

graph

Accepted Answer

Hello. In this video, we are going to be using the given graphs of the 1st and 2nd derivative of a function F, and we are going to use these graphs to determine which of the following options represent the shape of the original graph F of X. So, We are given F of X in the form of the red graph, and we are given the second derivative F of X in the form of the green graph. Now, in order to determine what the general shape of the original function F of X should be, we are going to mainly use the first derivative F of X. See, the important thing about F of X is that it graphs the values of all the slopes on the original function. And what that means is that for any part of FX that is below the X-axis, we are going to have a behavior of decreasing on the original graph. Furthermore, if the graph F of X is above the X-axis, then we are going to have an increasing value. So, let's go ahead and first start by labeling the X intercepts for the or for the function F X. Now, wherever we have an X intercept on FX, that means that the slope of F of X is 0. And what that means is that we can either have a local maximum or local minimum value. Let's go ahead and label these X intercepts as a point A and a point B. And what we are going to do is we are going to create a number line labeling point A and point B. Now, to the left of A, we are going from negative infinity and zeroing out towards A. But for this behavior of FX, since the derivative is below the X axis, that means the original function F of X is going to be decreasing in this region. Now, in the region between A and B, F of X is above the X-axis. And since F of X is above the X-axis, that means the original function F of X is increasing within this region. And finally, to the right of the point B. The graph F of X is below the X-axis, meaning that the original graph F of X is decreasing within this region. So we can go ahead and create a rough sketch of this. Let's go ahead. Oh, and one more thing, since the graph of F of X is switching from decreasing to increasing as we pass through A, this is indicating that we have a local minimum at the point A. Furthermore, since the graph is switching from increasing to decreasing as we pass through B, we have a local maximum. Now, let's go ahead and take a look at F of X. Now, for F of X, this tells us the concavity of the function. Now, notice that we have an X intercept at 0 for F of X. To the left of zero we have that the double derivative is above the x axis, and this means that the graph is going to be. Or the concavity is switching as we're passing through zero, since F of X is above the X-axis to the left, and is below the X-axis to the right, and that means that the 0.0 for the function F of X is going to be an inflection point. So let's go ahead and first draw the local maximum and local minimums. We can estimate for the point A to be roughly here, and the point B to be here. Furthermore, the graph is going to switch concavity as we pass to the 0.0 at the origin. Now, based off the sign chart, we are going to decrease towards A. So the graph will exhibit some behavior as the following. Then we are going to increase towards B while passing through zero, and notice that as you pass through zero, the concavity of the graph switches. And finally, as you pass through B, the graph is going to decrease. So this is going to be the rough shape of the original function F of X, and notice that option B matches the shape. Therefore, B is going to be the solution to this problem. So I hope this video helps you in understanding on how to approach this problem, and I will go ahead and see you all in the next video.

Each of Exercises 89–92 shows the graphs of the first and second derivatives of a function y=f(x). Copy the picture and add to it a sketch of the approximate graph of f, given that the graph passes through the point P.

Key Concepts

First Derivative

Second Derivative

Graphical Interpretation

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