Question

At what values of x does f have extrema.
Also, what intervals is f increasing? Decreasing?
At what values or x does f have inflection points?
On what values of x is f concave up? down?

Answer 1

f'=0 when x=-4,0,2,4

so looks before and after each of the above critical numbers to see if f is decreasing or increasing

intervals of increase: (-4,0) U (2,4)

intervals of decrease: (-inf,-4) U (0,2) U (4, inf)

f has max occurring at x=0 and x=4

f has min occurring at x=-4 and x=2

the function is concave up on intervals: (-inf,-3) U (1,3)

the function is concave down on intervals: (-3,1) U (3,inf)

inflection points are (-3,f(-3)) , (1, f(1)), (3,f(3))

Answer 2

The graph i attached was that of f prime, that is the graph of the derivative of f

Answer 3

ok well i did the above under the assumption that the graph was f lol

Answer 4

f'=0 when it cross the x axis so those are critical numbers

Answer 5

could you please do it for me?? please, i am sorry i forgot to mention that it was the graph of the derivaitve

Answer 6

f'=0 when x=-5,-3,1,3,5

so these are critical numbers these are where the extremas will occur

Answer 7

what about intervals where f is inceasing or decreasing

Answer 8

since f'>0 before x=-5, then it is increasing on (-inf,-5)
since f'<0 after x=-5 and f'<0 before x=-3, then it is decreasing on (-5,-3)
since f'>0 after x=-3 and f'>0 before x=1, then it is increasing on (-3,1)
since f'<0 after x=1 and f'<0 before x=3, then it is decreasing on (1,3)
since f'>0 after x=3 and f'>0 before x=5, then it is increasing on (3,5)
since f'<0 after x=5, then it is decreasing on (5,inf)

Answer 9

the inflection points will happen between the critical numbers

Answer 10

how abot concavity?

Answer 11

one sec lets talk about the the inflections first
or you can say where f''=0
that is where f' has horizontal tangents
so where do you see where f' has horizontal tangents?

Answer 12

that should be at x=-4, x=0, x=2 and x=4

Answer 13

very good
so inflection points are (-4,f(-4)), (0,f(0)), (2,f(2)),(4,f(4))

Answer 14

so, thiese are point where concavity changes, but....

Answer 15

so now lets think about how to determine where f is concave up and concave down

if we are looking at f we would be looking for U or n
U is where f would be concave up
n is where f would be concave down

so looking at that U it is going from decreasing to increasing

so looking at that n it is going from increasing to decreasing

so where do we have that it switches from decreasing to increasing
lets look at the above intervals that we have that it switches from
(-5,-3) to (-3,1)
and
(1,3) to (3,5)

so what inflection point is between (-5,1)
and what inflection point is between (1,5)

Answer 16

-4
and 4

Answer 17

sorry i couldn't find the problem
i was like omg i cant find mathdragon lol

Answer 18

lol

Answer 19

right so that means it is concave up on (-4,0) U (4,inf)
i determine these intervals because the -4 was in the first interval of (-5,1)
and 4 was in the last interval of (1,5)

Answer 20

does this make sense so far?

Answer 21

yes

Answer 22

ok so everywhere it wasn't concave up it is concave down right?
so we
have it is concave down on (-inf,-4) U (0,2) U (2,4)

Answer 23

right

Answer 24

but this would mean i made a mistake because this implies that 2 isn't an inflection point and it is

Answer 25

let me go to the restroom and i will be back

Answer 26

ok, hope everything comes out alright lol

Answer 27

I dont think x=0 is a crtical value of f prime

Answer 28

we have it concave up on (-inf,-4) since by looking at the graph we have increasing to decreasing

so that means everything else is easy since at -4,-2,0,2,4 the concavity switchs

so
(-4,-2) it is concave up
(-2,0) it is concave down
(0,2) it is concave up
(2,4) concave down
(4,inf) concave up

there i feel better about this

Answer 29

i feel 100% positive about my answer now :)

Answer 30

at x=0 both the f prime and f double prime are 0 right

Answer 31

no f' is not 0 at x=0
we are given the graph of f'
f' has 0 value when f' crosses the x-axis

Answer 32

just f'' has value 0 at x=0 since we can draw a horizontal tangent there

Answer 33

so f' is just positive there right

Answer 34

yes because it is above the x axis

Answer 35

hey i am going to draw the graph of f based on the info we got and then i will post it and i want you to check it for me, is that cool?

Answer 36

k

Answer 37

Final product, my bad if its a little sloppy lol

Answer 38

i don't see anything wrong with your graph

Answer 39

when i drew f' off your f we got the same f' we started with so gj

Answer 40

wish i could give you triple medals for helping me out all this time, thanks

Answer 41

np
looking off f is much easier than looking off f' because we are use to it
i had to think alittle for looking off f' instead