Use the following graphs to identify the points on the interva...

Question

Use the following graphs to identify the points on the interval [a, b] at which local and absolute extreme values occur.

Accepted Answer

Hi, everyone. Let's take a look at this practice problem. This problem says examine the function GFX plotted on the closed interval from -3.4 to 3.2. Identify the X values where the function attains its local and absolute extrema. Below the problem, we're given our graph of GFX, and our function GFX is increasing on the interval from -3.4 to -1.9, is decreasing on the interval from -1.9. To -1.2, it is increasing again on the interval from -1.2 to 0.6, and it's decreasing on the interval from 0.6 to 3.2. Now, I asked to identify the X value where a function attains local and absolute Xtrema. So we're gonna start off by looking at the local maxima. So here we need to identify X values, where our function is changing from increasing to decreasing. And we actually see that there are two points at which this occurs. This occurs at X equal to minus 1.9 and X equal to 0.6. For both of these uh X values, our function is increasing just to the left of them, and it's decreasing just to the right of them, right of them. So next we're going to look for local minima. So here we're gonna be looking for X values. That We are located where our function is changing from decreasing to increasing. And here we only have one point that satisfies that condition, and that is at X equal to minus 1.2. So now we need to identify our absolute maximum and absolute minimum. So we'll start off by looking at the absolute maximum. And so this is going to be the highest value that our function G of X actually obtains. And if we look at our graph, the highest value that our function obtains occurs when X is equal to -1.9. So our absolute maximum is located at X equal to -1.9. And the absolute minimum occurs when our function reaches its lowest value over our interval. And if we look at our graph of our function, we see that the lowest point occurs at the very end point of X equal to 3.2. So that is our absolute minimum. So, just to recap, we found that we have local maxima at X equal to -1.9 and X equal to 0.6. We have a local minimum located at X equal to -1.2. We have an absolute maximum located at X equal to -1.9, and we have an absolute minimum located at X equal to 3.2. So I hope that this explanation has been useful and I'll see you in the next video.

Use the following graphs to identify the points on the interval [a, b] at which local and absolute extreme values occur. <IMAGE>

Key Concepts

Local Extrema

Absolute Extrema

Critical Points

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