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University of California, Riverside

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Problem 27

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)=x^{4}-8 x^{2}+16

$$

Answer

See the graph

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## Discussion

## Video Transcript

Okay. Now for problem Time seven function. Thus is FX, which is defined as X cube minus eight X squared 16. Now, in part A. We need to take the trip to that which is four X cubed, minus 16 axe. And we let this to be positive and a person simply by thes thesis expression, excuse or axe, Times X plus two times X minus two. Bigger than zero. So to solve this inequality, we need to consider Ah, First we need to consider two conditions. That is, when all these expressions are positive so that we definitely will have a positive number on a different side of already in quality now to make, um yeah, so that gives over. So if all the expressions are positive, then that will be bust. Three inequalities persist for axe is positive that experts to is positive than X minus two. It's positive. And combine these 23 inequalities we can have our X uh, yeah, so we can have our rich bird acts, which is bigger than two. Now, the second condition we need to consider is, um well, two off the that is, too. Uh, three are negative now to do that That's first consider or ex smaller than zero and access plus two smaller than zero and X minus two has to be positive. Or we can. We can have four acts smaller than zero and acts less too bigger than zero and X minus two smaller than zero or our nest set of inequalities four x eager that zero and access to smaller than zero and exploit us to swell honor than zero Now combined. These combined these three conditions so Florida first condition. I combine these two by combining the these three inequalities we have, Empty said. That means there's no solution to this three in quality inequalities. Now, for the second set off in qualities, we do have a solution. I will skip the process here. It's our inequality for arrange for acts will be naked 2 to 0. Now the second condition. We also give us an empty set, so the last thing is to combine this result and this result. So that means our acts can be where the increasing devil can keep you from naked 2 to 0 you did due to infinity and on this interval or function will be increasing. Otherwise, from negative infinity to connect it to Union 0 to 2. All the depressing AL for RC South Park Beat for part B. Um, again, we have to solve the the grav off dysfunction just approximately. So we have connected to zero and two. Our function will be from Mecca to from negative infinity from negative Infinity connected to will be decreasing and from next 2 to 0 will be increasing and from 0 to 2 will be decreasing. And from 0 to 2, infinity will be increasing. So our local extreme, these three points that is happening to F zero and after two will be the three nobody streams. And for the absolute, um, for the absolute extreme, we need to check because next to and the two will be will be the two local minimum. You need to check whether which one will be bigger. It's like we shed f addicted to and after two, so the function value will be zero. Both of them will be due. So these two will be our absolute stream, both of to our absolute extreme Sorry

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