a. Find the open intervals on which the function is increasing and decreasing. b. Identify the function's local and absolute extreme values, if any, saying where they occur. $$ g(x)=x \sqrt{8-x^{2}} $$

Okay, so four problem 33 function is Jack's defined as X Times square root off eight minus eight x x squared. Okay, now, the first part, we need to take the degree of two. Um, yeah, but before we before we taking people, we take the three of tiff. I want to emphasize the demonic domain of dysfunction. So the thing we need to notice is dead because eight minus and minus X squared is under the square root. That means 80 miles squared of eight minus x squared has to be bigger or equal to zero. Okay. And under that condition, then we have X squared a smaller or equal to eight. And that gives our domain, which is from two times square root, too. Too connected to times square root, too. Okay, Now we take it to rib, too. G prime of X is defined. You pre apply the product rule to G X that we have square root off eight minus x squared plus 11 over two times eight minus squared, minus x x squared times. Remember, the change will have to take it to realty off the inside function, which is negative to x and times X. Okay, now we simplify. It is expression. Now we have square root of eight minus x x squared, minus minus, X squared, divided by aid miners square root eight minus X squared, minus X squared. Okay, now, because now we learned the square root off the miners. X squared is in the denominator. So in order to make this to Rubio, satisfied it. In order to make this derivative to be to exist, we have to make X. It's not equal to plus or minus two times Square with two. Because he actually is equal to either too square to times Square to than our denominator will be zero then this does This doesn't make sense. So this is our This is one more condition on for our interval. Oh, we have equality here. But after taking out the rib to we don't have the equality anymore. So that his ex speaker will you go then connected to Times Square to smaller than to Times Square to so this cellar domain Florida derivative. Okay, Now, um, the next thing is to make this that directive to be positive. You want to biased on to solve the inequality to see See the solution now. So this is our directive with that is to be positive and we multiply square with eight minus x squared on both sides. So we will have eight minus X squared, minus X squared on 11 side which you speaker than zero. So Dad gives aid is bigger than two x squared. So for his bigger than X squared, then that means eggs is in interval from negative to to to. So when exiting this interval, our function would be increasing. Otherwise, when exits from negative to times square to to negative too. Union 22 to Times Square would help too. On this interval, our function will be secrecy. Okay, Now the next part we need to find a local extreme, um is quite clear. If we observe from our inter boast and effective too and f two will be too local extremes for dysfunction. And for the episode absolute extreme, we need to check the end points. Remember 40 end points we have square root off, eh? Which is the same as the, um to Times Square with us too. And this is equal to the square root of eight and which is equal to zero. So these two end points will not will not be the be the absolute extreme. And so the only the only possible absolutely extreme are these two is to get this to function values which is f l connected to which is four naked, poor and never to which is for these two are absolute value. Sorry, absolute extreme better.

## Discussion

## Video Transcript

Okay, so four problem 33 function is Jack's defined as X Times square root off eight minus eight x x squared. Okay, now, the first part, we need to take the degree of two. Um, yeah, but before we before we taking people, we take the three of tiff. I want to emphasize the demonic domain of dysfunction. So the thing we need to notice is dead because eight minus and minus X squared is under the square root. That means 80 miles squared of eight minus x squared has to be bigger or equal to zero. Okay. And under that condition, then we have X squared a smaller or equal to eight. And that gives our domain, which is from two times square root, too. Too connected to times square root, too. Okay, Now we take it to rib, too. G prime of X is defined. You pre apply the product rule to G X that we have square root off eight minus x squared plus 11 over two times eight minus squared, minus x x squared times. Remember, the change will have to take it to realty off the inside function, which is negative to x and times X. Okay, now we simplify. It is expression. Now we have square root of eight minus x x squared, minus minus, X squared, divided by aid miners square root eight minus X squared, minus X squared. Okay, now, because now we learned the square root off the miners. X squared is in the denominator. So in order to make this to Rubio, satisfied it. In order to make this derivative to be to exist, we have to make X. It's not equal to plus or minus two times Square with two. Because he actually is equal to either too square to times Square to than our denominator will be zero then this does This doesn't make sense. So this is our This is one more condition on for our interval. Oh, we have equality here. But after taking out the rib to we don't have the equality anymore. So that his ex speaker will you go then connected to Times Square to smaller than to Times Square to so this cellar domain Florida derivative. Okay, Now, um, the next thing is to make this that directive to be positive. You want to biased on to solve the inequality to see See the solution now. So this is our directive with that is to be positive and we multiply square with eight minus x squared on both sides. So we will have eight minus X squared, minus X squared on 11 side which you speaker than zero. So Dad gives aid is bigger than two x squared. So for his bigger than X squared, then that means eggs is in interval from negative to to to. So when exiting this interval, our function would be increasing. Otherwise, when exits from negative to times square to to negative too. Union 22 to Times Square would help too. On this interval, our function will be secrecy. Okay, Now the next part we need to find a local extreme, um is quite clear. If we observe from our inter boast and effective too and f two will be too local extremes for dysfunction. And for the episode absolute extreme, we need to check the end points. Remember 40 end points we have square root off, eh? Which is the same as the, um to Times Square with us too. And this is equal to the square root of eight and which is equal to zero. So these two end points will not will not be the be the absolute extreme. And so the only the only possible absolutely extreme are these two is to get this to function values which is f l connected to which is four naked, poor and never to which is for these two are absolute value. Sorry, absolute extreme better.

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