Question

Function f(x) is not defined at x = −1. The derivative and second derivative of f(x) are
f'(x) = ((x+2)(x^2+x+2))/(x+1)

f''(x) = (-2(x+4))/(x+1)^4

Find:
a) all maxima and minima of f (only x − value can be found)
b) all open intervals on which f is increasing (give answer in interval form)
c) all open intervals on which f is concave downward (give answer in interval form)

Answer 1

When checking for f'(x) min max and f''(x) concavity and I am testing values, I put the values into the f'(x) and f''(x) functions right? I wouldn't use f(x)... Just trying to understand for my final!

Answer 2

Okay give me a couple mins almost done with my work!

Answer 3

I got only -2 and -1 as critical points, then tested -3 and -1.5 and -2 is a max. Didnt bother checking for right of -1 because it's not defined and can't be a min max right?

Answer 4

Oh I would need to still check for the increasing part of the question

Answer 5

No problem!

Answer 6

For some reason I got -3 being positive and -1.5 being negative hahahah, did I just screw my math up?

Answer 7

Mhmm!

Answer 8

Yes I get -2 and -1 as critical points

Answer 9

And -1 not being defined

Answer 10

http://tutorial.math.lamar.edu/Classes/CalcI/CriticalPoints.aspx

I'm super confused! I think they do come from f'(x)?

Answer 11

Yeah she's taught them as critical points. But that's just terminology preference I guess! So Don't we have to test for the left of -2, and in between -2 and -1? And I get positive when I test for -3 and negative when I test for -1.5... But that's a max not a min :(

Answer 12

Mhmm! I'm more worried about me getting a max for some reason

Answer 13

Yep yep

Answer 14

I get before -2 being positive and after -2 being negative.... I've checked my work idk. :-(

Answer 15

(-3+2)((-3)^2+(-3)+2)/(-3+1) = - / - = +

Answer 16

(-1.5+2)((-1.5)^2+(-1.5)+2)/(-1.5+1) = + / - = -

Answer 17

no problem at all :)

Answer 18

@amistre64 Please, help

Answer 19

what is it we are needing help with?

Answer 20

We were getting different ansers

Answer 21

answers*

Answer 22

f(x) is not defined at x = −1
f'(x) = ((x+2)(x^2+x+2))/(x+1)
f''(x) = (-2(x+4))/(x+1)^4

Find:
a) all maxima and minima of f (only x − value can be found)

f'(x) = 0 defines a horizontal slope, but not necessarily a min or max

min and max are defined by cavity ... or by observing if the slope changes sign.



b) all open intervals on which f is increasing (give answer in interval form)

an increaseing function has only a positive slope



c) all open intervals on which f is concave downward (give answer in interval form)

inflection points are required for this i beleive

Answer 23

f'(x) = ((x+2)(x^2+x+2))/(x+1) = 0 when the top is zero

x+2 = 0; or x^2+x+2 = 0
x=-2 1-4(2) give no real roots

Answer 24

I get -2 as a max, increasing on (-inf, -2) U (-1, inf), -4 as point to check for inflection

Answer 25

Is concave down or concave up when f''(x) > 0 and < 0?

Answer 26

-2 max, and increasing intervals i agree with, so it concavity we are having issues with?

Answer 27

Mhmm!

Answer 28

f''(x) = (-2(x+4))/(x+1)^4 = 0 or undefined

so -4 and -1 are critical points

Answer 29

Yep!

Answer 30

I get + for left of -4, - for in between -4 and -1, and - for after -1

Answer 31

the bottom is never negative so the top controls the sigh

-2(+) = cave down
-2(-) = cave up

x+4 > 0 when x > -4

x+4 < 0 when x < -4

Answer 32

So a + f''(x) is concave up and a negative is down?

Answer 33

is x^2 cave up?

f'' = 2 a positive value

Answer 34

Yep, yep it is

Answer 35

then we use that as our reference :) f'' + is cave up, f'' - is cave down

Answer 36

Thanks for the help!

Answer 37

remember x doesnt exist at -1, so keep that in mind when you do your interval of cave down

Answer 38

x > -4, except at x=-1

Answer 39

It's just concave down on (-4,-1)U(-1,inf) right? Just for certainty