Question

Find the values of x that give relative extrema for the function f(x) = 3x^5 - 5x^3.

Answer 1

can you take the derivative of this function?

Answer 2

yes its 15x^4 - 15x^2

Answer 3

great
solving for zero will give you some of the critical numbers
have you got that yet?

Answer 4

no but i think i did something wrong because as i equaled it to 0 i got x^3= x

Answer 5

well zero is one answer, but the best way to do it is to factor out 15x^2 giving
15x^2(x^2-1)=0
then the roots should be obvious

Answer 6

so is the relative maximum x=0 and relative minima x= plus or minus 1?

Answer 7

well we haven't checked which is which
at least I haven't, hold on...

Answer 8

the maximum occurs at x=-1
what made you think it was at zero?

Answer 9

how did you check which was a maxima and which was a minima?

Answer 10

i made a chart but i think i did it wrong

Answer 11

Breaking the solutions up into intervals and testing the value of the derivative in each interval can tell you which is which, is that what you meant by a chart?

Answer 12

yes exactly that

Answer 13

look at the space on the side of x<-1
take any point like x=-2 and plug it into the derivative f'(x)
if f'(-2)>0 then f(x) is increasing on that interval
if f'(-2)<0 then f(x) is decreasing on that interval
if the function changes from increasing to decreasing then it is a maximum
what did you get for that test?

Answer 14

mkay hold on my work is all over the place

Answer 15

if you can't find your work just check now
plug x=-2 into
f(x)=15x^4-15x^2

Answer 16

ok thank you so much for your help honestly i appreciate it because i have a calc packet to work on and i'm really quite clueless on more than half of it

Answer 17

anytime
let me know if you have any questions.

Answer 18

ok i got 809,100

Answer 19

lol
well the important thing is that it is >0 so we have the fact that f(x) is INCREASING on x<-1
so that is one interval taken care of
what did you get for a number in the next interval?
(-1,0)

Answer 20

and you should have gotten 180 for x=-2...
f'(-2)=15(-2)^4-15(-2)^2=240-60=180

Answer 21

Now that we know the first interval is decreasing pick any point from the next one
(-1,0)
say -1/2 or -1/4, something like that

Answer 22

ok i did 15x-2 and then placed the answer to the fourth power that's why

Answer 23

I see, it happens
so what about the next interval
pick a value for x and check f'(x) in the interval (-1,0)

Answer 24

ok i got -2.81

Answer 25

sounds reasonable
and what does that tell us about f(x) in that interval?

Answer 26

its decreasing

Answer 27

right, so we can make a chart...

Answer 28

so it looks like f(x) changed from increasing to decreasing at x=-1so what does that tell us about the point f(-1) ?

Answer 29

its the maximum

Answer 30

Good!
now test the next interval (0,1)

Answer 31

by the way, it's "A maximum" not necessarily the only one, we haven't tested yet to see if it is local or global.

Answer 32

ohh ok between 0 and 1 i got a negative

Answer 33

right, so we could see in our chart no change at x=0so we learn nothing. What about the last interval?