Problem

(a) Find the intervals of increase or decrease. …

03:13

Problem 24 Hard Difficulty

The graph of the derivative $f^{\prime}$ of a continuous function $f$ is shown.
(a) On what intervals is $f$ increasing or decreasing?
(b) At what values of $x$ does $f$ have a local maximum or minimum?
(c) On what intervals is $f$ concave upward or downward?
(d) State the $x$ -coordinate(s) of the point(s) of inflection.
(e) Assuming that $f(0)=0,$ sketch a graph of $f.$

Answer

A. $f(x)$ is increasing for $x \in(1,6) \cup(8, \infty)$
$f(x)$ is decreasing for $x \in(0,1) \cup(6,8)$
B. $f$ has a local maximum at 6 and local minima at 1 and $8 .$
C. $f$ is $\mathrm{CU}$ on $(0,2) \cup(3,5) \cup(7, \infty)$ and $\mathrm{CD}$ on $(2,3) \cup(5,7)$
D. $f$ has inflection points at $x=2,3,5$ and 7
E. SEE GRAPH

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Video Transcript

suffer on the given breath. We know that Ah, if prime is positive on, uh, these some intervals and these negative on the other intervals. So f is increasing home 1 to 6 and union with a tree infinity, and that is decreasing. 00 to 1. Union weighs 6 to 8. That means we have a local mix at X equals 26 And there we have a local Milliman. So I have to Local minimums at X equals two wind exceed coast rate. Now for the inflection part, we know that if prime f prime misty crazy. So if my is increasing phone 0 to 2 on the Ruutel five and all seven to Infinity, that means it's come that every is concave up. Because if if prime is increasing, that means the second the purity of his positive this country. Bob and F primes Deke raising long 2 to 3 union with 5 to 7 sitcom kept on. And let me is we have some infection point. Um at X equals 2 to 35 and seven because at each point, contribute ease are changing. So can um, GREss America sketch dysfunctional coordinate. Firstly, Laboard's off the intervals for one. The 12 um three, 45678 Okay, so if wrong 0 to 1, it's conch, A iced tea crazy in conclave town. So it looks like this. So there's a creating a point. No cold medium. No, it's con cave down and the increasing. Now we have a inflection point at X equals two. That means that can carry it. Is that different? Um, there is increasing. Come, keep up that you use this eso at X equals three. There's another inflection point that means the continuities there continue t will be different. It's known from 3 to 4 is increasing and can keep up. And, uh, from photo foot, actually, from 3 to 5 from 3 to 5, it's increasing in a conclave up. So we can just direct the graph this No, it it exit costs of five years. And now the inflection points in the from 5 to 6. It's increasing the contest. Um, and at six, we have a quick appoint a local man, Timon No, from 6 to 7. He crazy and calm down. Now, if seven continued its infection point, so from seven to hate, it's decreasing in the conclave. Up now. Eight. Yes. Ah, it's a local. So from eight to infinity, we have, uh, can keep popping increasing. It looks like it's

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