Identifying Extrema

In Exerc...

Question

Identifying Extrema

In Exercises 19–40:

a. Find the open intervals on which the function is increasing and those on which it is decreasing.

f(r) = 3r³ + 16r

Accepted Answer

Welcome back, everyone. In this problem, we want to determine the intervals where the function G of X equals 5x cubed minus 9X2 is increasing or decreasing. Now to figure out what those intervals are, let's try to find the critical points of our function and then choose parts within the intervals between those critical points to see if they are increasing or decreasing. Now what do we know about the critical point of a function? Well, at the critical point. The first derivative of the function is equal to 0. So we need to differentiate G of X, set it to 0, and solve for X. Now, the first derivative of G of X, if we go ahead, the derivative of 5 X cubed is going to be 15 X2 and of -9X2d will be -18X. So now if we set that to 0, OK, let me put that in red here just to show that we're setting it to 0. Then we should find that we get 15 X squared to be equal to 18 X, OK? Or rather, we could factor out 3 X from both terms to get 3 X multiplied by 5 X minus 6 to be equal to 0. And if that's the case, then one of these must be equal to 0. So this implies that either 3 X equals 0, which would make X equals 0, or 5 X minus 6 would be equal to 0, which means X would be equal to 65. So X equals 0 and X equals 65 are the critical points of our function. Now, how do we know which interval is increasing or decreasing? Well, based on these critical points, here we'll end up with 3 intervals, OK? So now if we go from negative infinity to positive infinity and put our critical points on our number line. Then notice here that we have our interval from negative infinity to 0. We have one from 0 to 65, and then we have one from 6/5s to infinity. So now, let's test where it's increasing or decreasing by figuring out the nature of the first derivative inside any of these intervals, because if it is increasing, our first derivative will be positive, but if it's decreasing, our first derivative will be negative. So now, let me put this in red here, testing our intervals. Read that properly. Testing intervals, OK. Then We can test our interval for. Between negative infinity to zero, open and closed parenthesis, OK. Let me make some space here. Uh, we can test it for 0 to 6/5 open parentheses. And then we can test it from 6/5 to infinity open parentheses. So now, from negative infinity to zero, let's just pick a value. Let's say that X equals -1. At X equals -1, OK, then our first derivative. Is going to be equal to 15 multiplied by 12 minus 18 multiplied by -1. 1 squared is 1, -18 multiplied by -1 is 18 and 15 + 18 equals 33. Our first derivative here is greater than 0, so this tells us that our function is increasing on the interval from negative infinity to zero, open and closed parentheses. Now how about from 0 to 6/5? Well, let's take x to be equal to 1. At X equals 1, our first derivative will be 15 multiplied by 1 minus 18 multiplied by 1. That's 15 minus 18 which equals -3, which is less than 0, is negative, so that tells us that our function is decreasing on this interval. Finally, from the interval from 6/5 to infinity, uh, let's take X to be equal to 2. Then in that case our first derivative will be equal to 15 multiplied by 2 square minus 18 multiplied by 2. That's 60 minus 36, OK, which. Which equals 24 and 24 is greater than 0, so that means it will be increasing on this interval. Thus, what can we conclude? Well, based on our results, this tells us then that our function is increasing, OK? On the interval, open parentheses negative infinity, 0 closed parentheses union with open parentheses 65 to infinity closed parentheses while it is decreasing. An open parenthesis 065 closed parenthesis. Thanks a lot for watching everyone. I hope this video helped.

Identifying Extrema

In Exercises 19–40:

a. Find the open intervals on which the function is increasing and those on which it is decreasing.

f(r) = 3r³ + 16r

Key Concepts

Critical Points

First Derivative Test

Increasing and Decreasing Intervals

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