Finding Extrema from Graphs

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Question

Finding Extrema from Graphs

In Exercises 15–20, sketch the graph of each function and determine whether the function has any absolute extreme values on its domain. Explain how your answer is consistent with Theorem 1.

g(x) = {−x, 0 ≤ x < 1

x − 1, 1 ≤ x ≤ 2

Accepted Answer

Welcome back, everyone. In this problem, we want to graph and determine the absolute extreme values of the function F of X equals x minus 3 when X is greater than or equal to -3 but less than 9, and X minus 9 when X is greater than or equal to 9 but less than or equal to 15. And here we have a graph on which we can put our function. Let's focus on the first part of this problem. Let's graph our function, OK? Now notice that for our function it's a piecewise function made up of two linear parts and to graph a line all we need is 2 points. So let's find those points to graph each piece of our function, and then we'll try to figure out the absolute extreme values. Now for our first interval, where X is greater than or equal to -3 but less than 9, OK, we can take two possible values. First, let's take F of -3. Now, when X equals -3, then F of X will be 3 or 3 minus 3, which equals 0, OK? So this tells us that the 0.-30 is on our graph, OK? On the other hand, when X equals 9, then F of X will be equal to -9 minus 3, which equals -12. So this also tells us that 912, 9-12, sorry. Will be a point on or a line. So now, if we go ahead and plot our points, when X equals -3, F of X equals 0, and when X equals, sorry, when X equals 9, F of X is -12. So it would be somewhere right here. I know at that point, we'd have an open circle because remember, X never is actually equal to 9. It's less than 9, OK? So, X equals 9 is not included in that interval. Now that we have our points, we can plot our line, OK? Let's look at it for the next interval. That is where X is greater than or equal to 9, but less than or equal to 15. No, our function will be X minus 9, and again, we can use that to find our points. When X equals 9, OK, F of X will be 9 minus 9, which equals 0. So that means 90 will be on our graph. And when X equals 15, F of X will be 15 minus 9, which is 6. So that tells us 15 6 is on our graph. So let's go to the points 90 and 156, OK? And that will also be a solid uh dot or a field dot because X is equal to 1515 is included and now here we've drawn the next part of our function. So of course we know that the points here, the important points are 156909-12 and 30, and this is the graph of F of X. So we've graphed our function. No, we need to determine the absolute extreme values for our function. Well, let's start with our absolute maximum. Basically, our absolute maximum, let me put that in red here. The absolute maximum is going to be the highest Y value on our graph. Now, if we think about it here, on the interval from -3 to 9, sorry, FX. Well, sorry, we're talking about the maximum, my apologies. Our highest value here is where X equals 15 and where X equals 15, or y value is 6. So the absolute maximum value is at 6, sorry, of 15. Which equals 6. So 6 is the absolute maximum value. Now hobo are absolute minimum value. Well, again, that should be the lowest y value on our graph and if we think about it. When we are looking at this interval here, on the interval from X greater than equal to -3 but less than 9, F of X decreases from 0 to 12, but it never reaches to -12. Sorry, it decreases from 0 to -12 but never reaches the negative 12, and we can tell that by our open circuit, OK? And at X equals 9, F of 9 is actually 0. So the greatest lower bound of FFX is -12, but it's not attained within the interval. So that means there is no absolute minimum value. Thus, our absolute maximum is 6 at X equals 15 and no absolute minimum exists for F of X. Thanks a lot for watching everyone. I hope this video helped.

Finding Extrema from Graphs

In Exercises 15–20, sketch the graph of each function and determine whether the function has any absolute extreme values on its domain. Explain how your answer is consistent with Theorem 1.

g(x) = {−x, 0 ≤ x < 1
x − 1, 1 ≤ x ≤ 2

Key Concepts

Absolute Extrema

Piecewise Functions

Theorem 1 (Extreme Value Theorem)

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