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-7)(x+1)(x+5)\end{equation}

a) $x=1 \quad x=-2 \quad x=3$

b) $(-\infty,-2) \quad(-2,1) \quad(1,3) \quad(3, \infty)$

c) $x=-2 \quad x=1 \quad x=3$

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Numerade Educator

Campbell University

University of Michigan - Ann Arbor

University of Nottingham

all right. So here are derivative of our function is X minus one time's experts, too. Time's explains three. Okay. And so first of all, we want to find our critical points said these air X values in the domain of F crime. Where that primacy there. Zero undefined. So we see X is one is going to make a crime. It's your exodus Negative, too. Next is three to have three critical points and then we wanted to get the open intervals where f is increasing or decreasing. So that means we want the intervals. We're F primes, either positive or negative civil client clerk, critical points on the number line and we'LL look att f crime and so to the left of negative to this factor is going to be negative. This factor is going to be negative and this factor is going to be negative. So we have negative making negative. The firm is going to be negative less than negative, too. Betweennegative too. And one will Now this term is going to be positive, but these two are still negative. So it's going to be, ah, one positive into negatives. So that's going to make a net positive and then between one and three. These two terms, we're positive that this was still going to be negative. Negative. And then to the right of three, all three of these terms or community positive. So for F death is decreasing than increasing. Decreasing Increasing. So decreasing. Uh, negative infinity to negative to increasing cremated Teo One increasing from one, two, three in an increasing saying through your insanity. Okay. And then we can see where our local extreme are so at negative too. We're changing from decreasing to increasing. So we have a local men at X equals singers too. We have a local maximum one because we changed from being increasing to decreasing and then another local men that X equals three because we changed from being decreasing but increasing. So if you just want to visualize what's happening or function is behaving like this

Georgia Southern University