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Georgia Southern University

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Problem 28

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^{4}-4 x^{3}+4 x^{2}

$$

Answer

See the graph

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## Discussion

## Video Transcript

Okay, here we are. So we have gox. It's next on the fourth minutes for X cubed polis for X squared. And if we take the derivative to find the critical points before X cubed minus twelve. Ah x squared plus x. Okay. Now, if you want to know in G Prime Zero Well, let's set this equal to zero. We can factor out for X. I never slept with X squared minus three X plus two is zero. And actually this factors as well. So you have X minus two X minus one. So you see, our critical points are zero one and two so we can draw us a number line. You're one two. That's nice and factor so you can see where G prime is going to change signs. So g prime appear to elect of zero. All of these terms are negative. So you have the negative times negative times negative. Give me negative and then between zero and one. This is going to be positive, negative, negative. So that's going to be in that positive. And then between one and two this is going to be positive, positive negatives that think it is, and in greater than to All of these terms were positive. So she it's gonna be depressing from minus infinity two zero increasing from zero, increasing from one to two and then increasing again from two to infinity. And this shows us Where are critical points? Are we heard? Sorry. Are extreme. Are so we have a local max where Well, we're changing from increasing to decreasing at one and the value we displaying in one into G and we get, um, one. And then the local men's we have two of them were changed from decreasing to increasing. So that's zero zero and two zero the value of zero. You can see all right. And so actually the absolute minimum. So this is a degree for you all know me also. It's going to be coming down and then up and then down. No. So I see that at zero and two, we have an absolute man. It's their absolute man is zero and it occurs at X equals Europe in X equals two. There's no local Max because the function is going off to infinity, to the left, into the right