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University of California, Riverside

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Problem 35

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.

$$

f(x)=\frac{x^{2}-3}{x-2}, \quad x \neq 2

$$

Answer

See the graph

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## Discussion

## Video Transcript

Kate. So four problem 35 function Keeping to us is a fraction with, um, that squared minus three over X minus two where X is not equal to two. So for the first part again, we need to take the derivative of dysfunction now because he's a fraction we need to apply the questions. Will. I'll write down the answer directly because it's just a a bunch of calculations that ihsaa X minus two squared in the denominator and X squared. Sorry, X squared minus or eggs plus three on the numerator. And we let this fraction to be positive. Now, the next thing is to solve this inequality. Now, remember, dead at all, we don't We don't allow X to be able to to sew the denominator. We'll always be positive. So the only thing we need to do is to require the numerator to be positive. And we simply by this we will get X minus one and at times X minus three on the inside. And by solving this inequality at one and three. And the rebels should be looked like this. What? Okay, so this is this our Paraiba. So that means when next is in the interval from naked 72 1 Union 32 halls to the infinity Dysfunction will be increasing. Otherwise, function will be depressing. Okay, so the next thing we need to do is to find a local extreme value. Now, if we'd draw the graph of dysfunction approximately, we can't find it. Just like a something like this. With this street is a straight line to be ex people, X equals two. Yeah, X equals two. And our function will be looked like this. Well, some here and sort of like this. Now, these two points will be our local extreme, which is F one F one and left. Three? Well, actually, yeah. So this one, this is three and our local, our local maximum and local minimum A three point. You'd be local Maximum on that three should be no called local minimum. And both of these are local extremes. Value loco extreme. And yes. Okay. Now to 40. Absolute extreme. We can find that there is no lower bound and no upper bound for dysfunction. So there's no absolute extreme values

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