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University of California, Riverside

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Problem 3

Answer the following questions about the functions whose derivatives are given:

\begin{equation}\begin{array}{l}{\text { a. What are the critical points of } f ?} \\ {\text { b. On what open intervals is } f \text { increasing or decreasing? }} \\ {\text { c. At what points, if any, does } f \text { assume local maximum and }} \\ \quad {\text { minimum values? }}\end{array}\end{equation}

\begin{equation}f^{\prime}(x)=(x-1)^{2}(x+2)\end{equation}

Answer

$$

\begin{array}{l}{\text { (a) Critical points } x=-2,1 ](b)[\text { see the graph }]} \\ {(c)[\text { No local maximum, local minimum at } x=-2}\end{array}

$$

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## Discussion

## Video Transcript

kept. So four problem. Three. All right, we have derivative off a function which is X minus one squared times X plus two. So similarly, uh, 1st 1st part, we need to find the, uh, critical points. So excuse me. That is to let this derivative to be equal to zero. Now, the solution will be X. What next? A sequel to one and when next is equal to negative too. Okay. No, To find the opening a rose for increasing and decreasing, we need to let the due to to be pause, do first. Now to solve this, um, to solve his inequality. The first thing we need to observe is dead X minus one squared will always be the negative. So that means we don't allow acts to be one. And at the same time, you should add in. And so, at the same time, we need to make experts to haunted. So that is, that gives the interval, which is from, uh let's see. Yeah, pushes for elective too. 21 Union one to infinity. And on this interval, the function will increase. And other than that, we will have the interval for decreasing, which is from negative. Infinity to elective too. Okay, now, Parsi. Now what are we going to do is to find a local maximum local animals. So it's quite simple. We just need to find We just need to check the locally streak. Lovely stream points. But the thing to notice instead, one is not. It's not a loco extreme. The reason is because we just exclude one from our increasing interval. And that means when x x equal to one, nothing will happen because he will keep increasing. So if one is not a local extreme, but have connected to will be a no Cole. This should be a local mutual okay and correspondent in there. There is no local man in Mexico.

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