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.
a. $(-\infty, 0) \quad(0,4) \quad(4,5)$
b. $(0,0)\quad(5,0); (4,16)$
all right here. You know G of X equals X squared Times Square ead of five minus six. So let's go and take the derivative because we want the critical point says to meet two X spirit of five minus X plus X squared over, actually minus because of the chain rule Teo Square to five month sex. The derivative of skirt. If I My sex is one over two square to find money, sex times negative one and we can simplify this a good bit. So this is if I multiply the top modern by, say, too square to find my sex to get a common denominator We'LL have for x times five minus x minus X squared all over too square to five minus x And so, if I want to you prime to be zero than I want Kayla's multiplies out. This is twenty x minus for X squared. Minus X squared is your, uh oh, this is twenty X minus five X squared, like in factor out of five ex. And when I was left with for ah minus X so X is zero R X is for and then also G prime can be un defiant. And that happens. The Nexus five. Okay, so critical points at zero for and fast. So let's show where she is increasing and decreasing. So have zero for five. So put it like a g prime. Okay, so there's gonna be some point where G prime is on to find right. And that's going to be how when X is greater than five. And also ji is going to be on to find there. So, undersigned over here, uh, the nexus is less than zero cheap crime. So in the new are just looking the numerator here, the denominators always going to be positive. Jeff, if X is less than zero, this is going to be negative. But this is going to be positive. So it's going to be negative. And then between zero and for this is going to be negative, huh? Sorry, this can be positive, and this is still going to be positive. And then between four and five, this is going to be negative. That's going to be positives. That b negative g This decrease saying increasing, decreasing and person to find over here, so decreasing between negative infinity and zero increasing between zero and four and then decreasing between four and five. And so good. We have local men's well, we can include five is a local men and zero is a local men. So zero zero zero gin's your refugee and then five. We also get zero for our value. So local men's of zero at X equals zero. Technicals five. And actually, this is going to be our absolute men. And that's easy to see because this is always greater than or equal to zero. So if we reached zero, that has to be the absolute men. And then we also have this local Max at four with the value of sixteen man Lucy. Yes, it does actually are absolute Max as well, because it is a unique point. Um, where? Yeah, it's a unique, absent maximums. This is actually, um, we actually have a actually, no, we don't have an absolute maximum. Okay, so we don't have the absolute maximum because, well, this function is decreasing from infinity. This way. So it's going off to infinity. Esos excuse to mice Infinity. This function is going to infinity, so no absolute maximum. But we do have an absolute minimum. The zero occurring it zero in five