Estimate the open intervals on which the function y = ƒ(𝓍) is...

Question

Estimate the open intervals on which the function y = ƒ(𝓍) is

a. increasing.

b. decreasing.

c. Use the given graph of ƒ' to indicate where any local extreme

values of the function occur, and whether each extreme

is a relative maximum or minimum.

Accepted Answer

Hello, in this video, we are going to be using the given derivative function in order to determine intervals of increasing and decreasing and local extrema. So, the problem says that based on the graph of J X, determine where the function JX has local extrema and classify them as either a relative maximum or minimum. We also want to determine the intervals where J of X is increasing or decreasing. So an important thing to note here is that the graph that is given to us is the graph of the derivative of X. What this means is that this graph is mapping all the values of the slopes of the original function J of X. The important thing to know here is that when J X is above the X axis, or in other words, it's greater than 0, the original function J of X is increasing within that region. Furthermore, if the derivative function is below the X axis, or it is less than 0, then J X is decreasing. Within that region So before we determine these intervals, let's just quickly determine the domain of the function. Now, the smallest X value that the graph exists at is that X is equal to -5, and the largest X value that the graph exists at is going to be at the value X is equal to 10. Furthermore, we have a discontinuity located at X is equal to 1. We also have an intercept of the function, J X located at X is equal to -2. So, the domain of this function is going to be from -5 to 10. Let's go ahead and start by identifying the intervals of increasing. Now, notice that from -5 to -2, the derivative is above the x-axis. Furthermore, on the interval from 1 to 10, the derivative is also above the X-axis. What this means is that the original function J X will be increasing on the interval from -5 to -2. And it is increasing on the interval from 1 to 10. So we can represent that as -5, -2, union 1 to 10. This is going to be the interval of increasing. Now let's go ahead and determine the intervals of decreasing. Notice that the given derivative function is below the x-axis on the interval from -2 to 1. This is the only part of the derivative that is below the X axis, and that means that the original function JFX is going to be decreasing on the interval from -2 to 1. So this is going to be the first answer for this question. Now, what about the local maximum or local minimum? While a local maximum and local minimum can only exist where the graph is continuous, since we have a jump discontinuity located at X is equal to 1, we know that there is not going to be a local maximum or minimum located within that region. But what about around -2? Well, since the graph is switching from increasing to decreasing, as it passes through the value of -2, what that indicates is that we have a local maximum. So, we have a local maximum located at the value X is equal to -2, and that is going to be the only local extrema for the original function. 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.

Estimate the open intervals on which the function y = ƒ(𝓍) is

a. increasing.
b. decreasing.
c. Use the given graph of ƒ' to indicate where any local extreme
values of the function occur, and whether each extreme
is a relative maximum or minimum.
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Key Concepts

Derivative and its Sign

Critical Points

First Derivative Test

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