Increasing and decreasing functions. Find the intervals on whi...

Question

Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.

f(x) = x²√(9 - x²) on (-3,3)

Accepted Answer

Welcome back, everyone. In this problem, we want to find the intervals of increase and decrease for the function H of X equals 3x2 multiplied by route 16 minus X2red on the interval -44. A says it's increasing on 04 root 2/3 and decreasing on 4 root 2/3 4. B increasing on 44 root 2/3 and decreasing on -4 root 2/3 0. C says it's increasing on 44 root 2/3 union with 0 for root 2/3 and decreasing on 4 root 2/30 union with 4 root 2/3 4. And the DA says it's increasing a negative 4 root 2/30 union with 4 root 2/34, and decreasing a 44 root 2/3 union with 04 root 2/3. Now, to find the intervals of increase and decrease, let's first differentiate H of X. Now, here, notice that HF X is a product and to differentiate a product, we can use the product rule of differentiation. The product rule basically tells us that if Y equals UV where U and V are functions of X, then the derivative of Y with respect to X will be equal to U V plus UV where U and V are the derivatives of U and V respectively. So in our in in this problem for age of X, if we let you be equal to 3 X squared. And V be equal to route 16 minus X2. Then the derivative of U will be equal to 6 X, OK? And the derivative of V by the chain rule is going to be equal to negative X sorry, divided by route 16 minus X2, OK? Know that we have our derivatives, then we can go ahead and apply the product rule. Because no, OK, this implies that the derivative of H of X is going to be equal to. 6 X. 6 X multiplied by a root 16 minus X squared. OK. Plus 3 eggs, OK. Multiplying here, so that's plus 3 X X squared, sorry, multiplied by negative X divided by root 16 minus X2. Now we can combine our terms here with our common uh uh by making route 16 minus X square, the common denominator, OK, multiplying through by our x terms here and in the interest of time when we do that, then eventually we'll get the derivatives to be equal to x divided by route 16 minus X2 multiplied by 96 minus 9 X2. This is the derivative of H of X. Now that we have the derivative, we can start to think about our intervals. And our intervals, to find our intervals, we can use the critical points of our graph. Now, what do we know about our critical points? Well, at the critical points. And at the critical points, the derivative of our function will be equal to 0. Therefore, this implies then that we can set X divided by route 16 minus X2 multiplied by 96 minus 9 X2 to be equal to 0. And in this case, OK, then. This implies that x divided by root 16 minus X2 will be equal to 0. And 96 minus 9X2 will be equal to 0. Now in the first case in in this scenario if we multiply both sides by route 16 minus X2d, then we'll get X to be equal to 0. And in our second scenario 9 x2d will be equal to 96, which means that X will be equal to. The square root, OK, square root of 96 divided by 9 or 32/3, OK? So this is our value for X. Now if we think about it here, we could factor out, uh, we could factor out 16, OK? And that would give us X to be equal to plus minus 4 multiplied by the square root of 2/3, which is going to be approximately plus R minus 3.27. But we'll keep it in this form for no, just for accuracy, OK? Thus The critical points, the critical points, OK. Are going to be X equals 0 and X equals plus R minus 4 root 2/3. Now what does that mean? Well, if we think about it, OK, if we were to put this on a number line. OK, uh, let me, uh, I think this is, I think this is a good enough distance. Where we have halfway through, we have 0. Remember we are looking for our intervals between -4 to 4. So here we have -4 and then we have 4. We also know that our critical points are at X equals 0 and plus R minus 4 root 2/3, OK? So we can say that 1 will be here, -4 root 2/3, and then for root 2/3. Essentially, we have 4 intervals that we're considering. We're looking at the interval between -4 and -4 root 2/3, the one between -4 root 2/3 and 0. Between 0 and 4 root 2/3 and then between 4 root 2/3 and 4. So we need to use or we need to analyze the sign of H or the derivative of H on these intervals because remember, if the derivative of H is positive, that means our function is increasing and if the derivative is negative, that means the function is decreasing. So let's start to look at each of these intervals to help us figure out what's going on. And remember, Remember that the derivative, I'll just write it here for reference, OK, is equal to X divided by route 16 minus X2 multiplied by 96 minus 9 X2. OK? Just to, just to keep it handy for us to be aware. Now, let's start to look at each of these intervals, OK? And usually we would just pick a random point in the interval, but because our expression is, it's a little bit, uh, complicated, we can instead start to think about the science. That might make it easier to, to, to analyze, OK? So, let's first look at when X is in the interval -4. -4 root23s. Now what do you, what do you think is going to happen to our derivative? Well here OK. In this interval, we know that X essentially is going to be negative, OK? And if X is negative, OK, then we also know if X is negative, then our term X divided by root 16 minus X squad is going to be negative, OK? So X. Let me write it here. X is going to be negative and our next term 96 minus 9 X squared is also going to be negative, OK, because this value X is going to be uh X is going to X2 is less is going to be less than 323 X is going to be less than -4 uh root 2/3, OK, so in this case. This expression will be negative. This expression will also be negative. Thus that tells us H of X is going to be increasing. And why is it increasing? Because here a negative multiplied by a negative is going to be equal to a positive. OK? Does that make sense? So we can just kind of think about the signs as we go along to help us to figure out whether our function is increasing or decreasing. And I'll write that above it here negative, multiplied by negative equals positive. Now, for our next interval, OK, where X is in -4 root 2/3, 0, OK. Now, we can tell that X is still going to be negative, but no, 96 minus 9 X squared is going to be positive, OK? Because Xsquared. We're gonna have our term here and that factors very much into the sign of 96 minus 9 X 2. So in this case during or inside this interval here, that means H of X is going to be decreasing, OK? And we know it's decreasing because our first term is going to be negative, our second term is going to be positive, so our derivative is going to be negative, OK? Now let's consider what's happening in the next two intervals. No, for X in the interval 0 for root 2/3, OK. No, we know that X is going to be positive, OK? And our expression 96 minus 9X2 is also going to be positive. So since both of them are are positive, that means H of X is increasing. We'll have a positive term multiplied by a positive term. And now finally for X in the interval for root 2/3, 4, OK. X is going to be positive. R X is still positive, right? But no, our expression 9 X. sorry, 96 minus 9 X squared is going to be negative, OK? Because our, our X value here will be greater than 4 root 2/3 and 96 minus 9 multiplied that critical value squared is going to make it pos going to make it negative. So in this case, that means H of X is decreasing. We have a positive term multiplied by a negative term. So what can we conclude based on what we've noted here? Well, for the ones that were increasing, let's start there, OK. We can tell. Let's put that in red. That age of X is increasing. On -4, 4 root 2/3, and so uh our union with 04 root 2/3. While it is decreasing, OK. And decreasing on. -4 root 2/30 union with 4 root 2/3 4. Therefore, if we look back on our answer choices, that tells us that C is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.

f(x) = x²√(9 - x²) on (-3,3)

Key Concepts

Critical Points

First Derivative Test

Intervals of Increase and Decrease

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