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Georgia Southern University

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Problem 2

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)(x+2)\end{equation}

Answer

$$

\begin{array}{l}{\text { (a) Critical points } x=-2,1 ](b) \text { see the graph }} \\ {\text { (c) local maximum at } x=-2, \text { local minimum at } x=1}\end{array}

$$

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## Discussion

## Video Transcript

So you're giving the derivative of a function F Crime of X is explain this one times X plus two. And so let's find the critical points of that. Well, these are the values of X through its theft crime of X zero Cor, undefined. But this derivatives never going to be undefined Soon X equals one and X equals negative to our critical points who just plug in one a negative to those will be the values vex to make of prime zero. Okay, And then we want to know All right, what values is f increased, their decreasing. Well, that's just asking it. What values Andi or what Open intervals is f prime, positive or negative. So you know, the F prime will potentially change sign at the critical points. So put not just to you here and one here and no test f prime so that blogging something smaller the negative to this factor is negative. This factor will be also negative, said F prime will be positive. And then, if I plug in and value between you two and one, this factor will be negative, and this factor will be positive. So the drip it it'll connective. And then if X is bigger than one and bigger than to both of these factors were positive. And so what that means for Earth is that ofthis increasing here, Decreasing here, increasing here. So it's increasing on this interval. Negative infinity. It's a negative, too decreasing between aged two and one and then increasing between one and infinity. All right, And then at what points does f assume local maximum minimum guys So it looks like here have changes from being increasing to decreasing. So hey, we have a local max at X equals negative too. And then we have a local men because of changes from decreasing to increasing at X equals one. Then a pre just sort of visualize what that looks like. It's increasing and decreasing that increasing. So we have a local max in the community.