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.

f(x) = |x|, −1 < x < 2

Accepted Answer

Below there. Today we're going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Graph the function and determine the absolute extreme values of the function. F of X is equal to the absolute value of X plus 2, -2 is less than X and X is less than 1. Awesome. So it appears for this particular problem, we're ultimately trying to create a graph of the specific function of FFX, and we're also asked to determine the absolute extreme values of the function. So we're trying to determine if there is an absolute minimum and or maximum value. Awesome. So with that said, now that we know what we're ultimately trying to solve for, first off, in order to graph the function. F of X is equal to the absolute value of X + 2 over our designated interval, which is -2 is less than X and X is less than 1, and we're asked to determine its absolute extreme values. We need to follow these steps to help us solve for those answers. So our first step is we need to graph points. We need to determine graph points and calculate them, so we need to calculate our graph points. So, With that said, so 4 F of X is equal to the absolute value of X plus 2. The function is a V-shaped graph because of the absolute value term. So let us evaluate the function at a few key points within our given interval. So with that said, let us evaluate it at the following points. So we're gonna check, we're gonna basically pick out different values of X that fall within our interval. So let us start with F of -2, since -2 falls within our interval. So F of -2 is equal to the absolute value of -2, which anything within so anything that's inside of absolute value signs will become positive. So the absolute value of -2 + 2 will be equal to 4, which will give us an undefined answer. So I'll just say UD undefined. Answer. Let us also take, so technically, since we say that X is greater than -2, so that's why we have an undefined answer. So that's technically not within our interval. So looking at F of -1, so we're plugging in a value of -1 in 4 X. So instead of writing out the expression every single time, I'm just going to give you the evaluated function answer. So when we plug in a value of -1 into our function, we will find that F of -1 is going to be simply equal to 3. F of 0 is going to be equal to 2. Once again, we're plugging in the value of 0 in for X, and finally F of 1 will give us an answer of 3, which is also undefined. Fantastic. So, Because we have our interval, so X is greater than -2, so when we plug in a value -2, we get an answer of 4, which will give us undefined, and then if you plug in a value of 1, we will find that it's equal to 3, which will also give us an undefined answer. So, moving right along here. So now we need to determine the absolute extreme value. So the absolute minimum value occurs at And once again, this is our 2nd step, so I should make a note of that. So this is step 2, so now we're focusing our attention on determining the absolute extreme values. So let us note that the absolute minimum value occurs at X equals 0, so this is our absolute. Min, I'll call it a subscript min for short. So the absolute min occurs at X equals 0, where the function reaches its lowest value of 2, and the absolute maximum value is at the end points of the interval, which is approaching X equals -2. So let's make it another, so this is the absolute max. So it's going to be approaching X equals -2, and X equals 1. Fantastic. So, therefore, here, the function reaches 4 at X equals -2 and 3 at X equals 1, but there the function is undefined at these points, and because it's undefined at these points, there is no absolute maximum. So in conclusion, when we go ahead and use our graphing software or choice, or if we plot our graph by hand, we should produce the following graph. And as you can see, our final two or I should say final three answers are the absolute minima is at 0.2. And our absolute maximum, you need to know that there's no absolute maximum value. And then when we graph our V graph, we will see that our undefined points are indicated with hollow circles, which these when you probably might have heard them referred to as holes. And once again we have these values. At the end points of our V graph, because that's where it's undefined. And once again, the curve is represented in a solid black line, and this is once again our graph for the function F of X is equal to the absolute value of X plus 2. And that's it. We've officially solved for this problem. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

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.

f(x) = |x|, −1 < x < 2

Key Concepts

Absolute Extrema

Graphing Piecewise Functions

Theorem 1 (Extreme Value Theorem)

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