101. In Exercises 101 and 102, the graph of f' is given. Deter...

Question

101. In Exercises 101 and 102, the graph of f' is given. Determine x-values corresponding to local minima, local maxima, and inflection points for the graph of f.

Graph of f' showing local minima and maxima, with x-axis ranging from -4 to 4 and y-axis from -4 to 4.

Accepted Answer

Welcome back, everyone. In this problem, given the graph of a derivative of the function F X, identify the X values where F of X has a local maxima, and the local minima. And here we have a graph of our derivative of F of X. Now what do we know about derivatives that will help us to identify our local extremals, that is the local maxima and minima. Well, remember or recall, OK, that at the minimum point or at our local extremum, so let's make a note here, recall that at local extremum. OK, for our graph. Then the first derivative. Is going to be equal to 0. So that means if we can look at our graph, we'll be able to find our local extremal values at those points. So where is our graph equal to 0? Well, notice it is equal to 0 at X equals 2 X equals 1, and X equals 4. So now we know where those values are. All we need to do is to figure out then if they are maximum or minimum values. How can we tell? Well, we know that if it goes from positive to negative, then it will be a local maximum, and if it goes from negative to positive, it will be a local minimum. So what's happening at each of our points? Well, let's start with X equals -2. Notice that at X equals -2, or derivative changes from positive to negative, OK. So since it goes from positive to negative, that means that here X equals -2 has a local maximum. All right, it is a local maximum. Or rather, let's not say this, it corresponds, OK? It corresponds to a local maximum. I think I need to write this better. Let's take out the 8. Let's just say this value corresponds to a local maximum. OK. What else? Well, let's go to an X equals 1. Notice that at XC equals 1, our derivative goes from negative to positive. So in this case, because it goes from negative to positive, that means it is a local minimum, OK? And now finally at X equals 4 we can tell that it goes from positive to negative, so this will also be a local maximum. Therefore, from the graph of the derivative of F of X, we can tell. That our local maxima, OK. We have two local maximums. So our local maxima is at X equals -2, and X equals 4. While the local minimum, OK, one local minimum value here is going to be at X equals 1. Thanks a lot for watching everyone. I hope this video helped.

101. In Exercises 101 and 102, the graph of f' is given. Determine x-values corresponding to local minima, local maxima, and inflection points for the graph of f.

Key Concepts

Critical Points

Local Maxima and Minima

Inflection Points

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