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)=4 \sqrt{x}-x^{2}+3

$$

a. $-\infty$

b. max (1,6); min (0,3)

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Missouri State University

Campbell University

Harvey Mudd College

Idaho State University

very so here we have this function. G FX is for your necks minus X squared plus three. Let's take the derivative Get Teo over. Relax, friends. Two eggs. So if you want to keep trying to be zero, it's a critical points. We can rewrite G prime as two minus two acts rude eggs although a rude X that equals zero. And so we see what values of X we're going to give us zero Well, and the numerator we're just gonna get X equals one. But also, we noticed the G prime could be undefined when access here. So we get to critical points. X equals one in X equals zero. So if you draw us a number line put on there zero one And, of course, the domain of G. It has to be greater than equal to zero. So, actually, if we look at the values of t prime okay, two left, Well, it's going to undefined. And then between zero and one, this is going to be smaller than ones are going to minus something smaller than to, so that's going to be positive. And then when this is greater than one two months, something greater than to be negative. Okay, so then, G, it's going to be course on defined over here. And then it's going to be increasing from zero to one and decreasing from one to infinity Craig. And so it's important to know that our absolute max Sorry. OK, so let's let's talk first about local extremists. So we definitely have a local max and that one because the G changes from being increasing to decreasing and the value will display in one we get it looks like six. Okay. And so we also I mean so because G is to find a zero, we can say that we have a local men if we want. I mean, I don't think you have to say this, but, uh, it's sort of up tio up to debate whether or not this is actually local men, because this is the end point of the domain. But I mean, waken say it's a local man At zero, we're gonna value three. But we do need to realize that because our function pan is going to negative infinity, we do not have an absolute minimum, So the function is going toe. I kind of started here zero and go up increasing, then decrease down and keep going down to insanity. Uh, so you know, how do we justify that? What we just take the limit is excuse to infinity and this minus X squared terms going to be the most dominant term. Um, because of this is going I intentionally we get an intermediate form so we can have to write is a fraction still wakes up. You take the limit is ex ghost infinity of? So I'm gonna rewrite G of X as come for X minus X to the If I have plus three, next to the one half over twenty is going to be X to the one half. In that case, well, okay, so the degree of the top is greater than the decree of the bottoms is going to infinity. That's right. Mind's ability because it's minus X to the firehouse. There's absolutely nothing. That's the point. But there is an absolute maximum, and it is equal to six, and it occurs at X equals one and again this because it's the only ah local maximum and it's decreasing and decreasing to the left, to the right

Georgia Southern University