Problem

The graph of the derivative $f^{\prime}$ of a con…

04:20

Problem 23 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 $x \in(0,2) \cup(4,6) \cup(8, \infty)$
$f(x)$ is decreasing for $x \in(2,4) \cup(6,8)$
B. f has a local maximum at 2 and 6 and local minima at 4 and $8 .$
C. $f$ is $\mathrm{CU}$ on $(3,6) \cup(6, \infty)$ and $\mathrm{CD}$ on $(0,3)$
D. $f$ has an inflection point at $x=3$
E. SEE GRAPH

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Calculus 1 / AB

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Chapter 4

APPLICATIONS OF DIFFERENTIATION

Section 3

Derivatives and the Shapes of Graphs

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Video Transcript

from the given graph off the purity if off if we know the following information So party so increasing uh, interval will be wrong. 0 to 2 union with 4 to 6 and the union wave Hey, to Not because over on these intervals f prime Miss politics and for decreasing part It's from 2 to 4. Um and the union was 62 So from part A, we know there is a local. Yes, I'm local maximum look monuments. Local mexes at X equals 22 and X equals the six. No commitment is that X equals two for index equals Great, huh? No, for the posse, we are looking for the inflection concurrently, An inflection point. So, uh, from the graph, we know if prime used if prime mystique raising when 0 to 3 and there is increasing on 3 to 3 to six union always 6 to 9. So we need to exclude, um X equals a six year because And this points Not if it's not differential. Knowing eso if I've prime is decreasing. That means they have top a couple prime years this land zero So it's conclave conclave down and ah, if if prime is increasing so it can keep up. So we have lasting go where for one inflection points at X equals the three. So with this information, we can give a sketch for this function. So first relay boat off that intervals from 23 four, 26 and eight. Okay, so on the first interval, it's increasing and it's conclave down suits. You would be something like this and it's Deke Raising the conclave down and there's three is the inflection points, so you will change the concave. Biti's um, maybe you should put something like this. So at this point, X equal to three other communities will be different. Maybe something like this and the one X equals 24 its analogue greater point. So it has a horizontal tension line, the same SX e coast to So it's that the function starting crazy at this point. Okay, so we can see there's a trump at X equals 26 Damien's. There's a cast here. It's a special point. It's not differential anymore. Um, so you would be something like this. And again when X equals to eight. This Ah, quit your point. So it has a hard time. Potentially. So this is ah schedule

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