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.

a. Find the critical points of f and determine where f is increasing and where it is decreasing.

Accepted Answer

Hello. In this video, we are going to be determining the intervals of increasing and decreasing, and the critical points of the function F of X by using the given graph of the derivative F of X. So, let's go ahead and start by finding the critical points. Of F of X by using the graph of F of X, the derivative. Now, the critical points are points that are located on the original graph where the slope is zero. Now keep in mind that the derivative F of X maps all of the slopes. Of F of X. What this means is that FX has all the values of every slope at any point of F of X. Now, since the critical points are located where the slope is zero, what this means is that the critical points are going to act as the X intercepts to the derivative graph. Now, if we are taking a look at the given graph, we have 3 X intercepts in total. The first one is located at -2, the second one located at -1, and the third one located at 1. What this means is that we have 3 critical points to the graph of F of X. The critical points are located at X is equal to -2, -1, and 1, and this is going to be the first solution to this problem. Now what we want to do is we want to identify the intervals of increasing and decreasing. Now, in order to identify these intervals, there's two things we need to understand. The first thing being is that if the graph of F of X is increasing, That means that the slopes are positive, thus, the graph of F of X will be greater than 0 or above the X axis. And vice versa if F of X is decreasing. Then that means that the slopes are negative and the derivative F of X will be less than 0. So, what this means is that if we want to identify the intervals of increasing, we want to write the intervals of the given F function that are above the X-axis. Now the first interval that we see that is above the x axis is between 2 and -1. Since this portion of the graph of the derivative lies above the x axis, the first interval of increasing for F of X is going to be from -2 to -1. Now, we do have a 2nd interval of increase. There is a second interval in which F of X is above the X-axis, and that is going to be from -1 to positive infinity. So we have two intervals of increasing in total, from -2 to -1, union 1 to positive infinity. That is going to be the interval of increase. Now we need to do is what we need to do is identify the intervals of decreasing. Now, in order to identify the intervals of decreasing, we want to find the intervals of the graph in which the graph of F of X is below the X-axis. The first interval we can see is from negative infinity to -2. This whole interval is where the derivative lies below the X-axis, and that is going to be the first interval of decreasing for F of X. Then the second interval where the derivative lies below the X-axis is from -1 to 1. What this means is that we are going to union the interval of decreasing with -1 to 1. And these are going to be our intervals of increase and decrease to the function F of X, which will be the final 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.

Use the graphs of ƒ' and ƒ" to complete the following steps. <IMAGE>
a. Find the critical points of f and determine where f is increasing and where it is decreasing.

Key Concepts

Critical Points

Increasing and Decreasing Intervals

Second Derivative Test

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