Use the graphs of ƒ' and ƒ" to complete the following ste...

Question

Use the graphs of ƒ' and ƒ" to complete the following steps.

c. Determine where f has local maxima and minima.

Accepted Answer

Hello. In this video, we are going to be determining the local minimum and maximum values of F of X using the graph of its derivative. So, how do we start this type of problem? Well, first, we need to acknowledge the fact that we are given the graph of the derivative of F of X. The reason why this is important is because the graph of the derivative maps all the values of the slopes of the original function F of X. So, how do we use this information to find the local minimum and maximum values? Well, if we think about the original function F of X, the local minimum and maximum values are either located at the top of a mountain or at the bottom of a valley. This type of behavior indicates that the slope at these particular points are going to be zero. So, in order to identify the local minimum and maximum values, we first need to identify the critical points of F of X. Since the slopes of F of X have a slope of zero at the critical points, that means that for the function F of X, the derivative, we want to find the X intercepts of the derivative. The reason with this being is because at the X intercepts, the X intercepts indicate the fact that the original function has a slope of zero. The X intercepts of the derivative are located at -2. -1 and 1. So all three of these points have the potential to either be a maximum or minimum value. Now, how do we identify the difference between the maximum and minimum value? Well, for a maximum value, a maximum value occurs when the graph is increasing to a certain point of X and then decreases afterward. And for a minimum value. It is the opposite of a maximum. The graph will decrease to the left of the critical point and then increase to the right of the critical point. So, let's go ahead and start by observing the behaviors of the graph at the critical 0.-2. Now, since we are given the graph of the derivative, the graph of the derivative will indicate that the original function F of X is decreasing if the graph of the derivative is below the X-axis, and the original graph F of X will be increasing if the graph of the derivative is above the x-axis. If we are taking a look at the critical 0.-2 the derivative lies below the x axis when it is approaching 2, and when we are leaving to approaching -1, we are above the X axis. What this means is that for the function F of X. At the point of -2, the graph will be decreasing to the left of -2 and increasing to the right of -2. What this means is that this matches the behavior. Of a minimum value, so -2 will be a minimum value. Now, what about the value of -1? Well, at the critical point weight -1, we know that as we approach the point, the graph is going to be increasing. But when we are leaving -1, going to the right of -1, the derivative graph switches to be below the x axis, and that means that the original graph F of X will be decreasing in this region. Since the graph is increasing to the left of -1 and decreasing to the right of -1, that matches the characteristics of a maximum value. Now, we just need to check the values around 1. We know that to the left of 1, the derivative lies below the x axis, so the original graph F of X is decreasing in this region. And to the right of one, the derivative lies above the X-axis, which means that the original graph F of X will be increasing in this region. And since we are decreasing to the left of -1, a positive one, and we are increasing to the right of positive 1, this matches the characteristics of being a minimum value. So what this means is that at the values X is equal to -2, and X is equal to 1, we have a local minimum value. And at the point, the critical point, X is equal to -1, we have a local maximum. And these are going to be the solutions to this problem. So I hope this video helps you identifying local mens and maxes of the using the given derivative graph, and I hope to see you all in the next video.

Use the graphs of ƒ' and ƒ" to complete the following steps. <IMAGE>
c. Determine where f has local maxima and minima.

Key Concepts

Critical Points

First Derivative Test

Second Derivative Test

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