Question

if f'(x) = -(ln x -1)^2(sinx-2)(x-3), x>0, what is the relative minimum and maximum of f(x)?

Answer 1

i know to find the extrema i have to find the derivative and then set it equal to 0 but i get confused with ln

Answer 2

we can find the critical numbers of f
by setting f'=0 and solving for x
you have f' already...
so find when each of the factors equal 0
ln(x)-1=0
when x=?

sin(x)-2=0
never happens because sin(x) has range from -1 to 1
and 2 is certainly not in that set

x-3=0
when x=?

Answer 3

ooh so we dont have to find a derivative?

Answer 4

the derivative was already given

Answer 5

we can find where the relative min and max occur
but I don't know if I can help you find what they are
because f' doesn't look like an elementary integral

Answer 6

so would the relative minimum be at x = e?

Answer 7

oh ok!

Answer 8

to determine if there is a relative min at x=e
we must make sure the function goes from decreasing to increasing there
we will need to test the intervals around the critical numbers

Answer 9

ok do we plug it in to the function?

Answer 10

f'(0)=?
f'(2.01)=?
f'(4)=?

Answer 11

I just randomly chose a number to test from each interval

Answer 12

k!o

Answer 13

ok!

Answer 14

oop f'(0)=?
f'(2.5)=?
f'(4)=?

Answer 15

ignore the 2.01

Answer 16

so now do we test each number into the function?

Answer 17

you test the intervals around the critical numbers

Answer 18

the critical numbers were e and 3

Answer 19

so you need to test a number before e
test a number between e and 3
test a number after 3

Answer 20

you see i have already chosen numbers to plug in
but you do not have to choose the exact ones I chose
for example for the first interval I chose to evaluate f'(0)
second interval I chose to evaluate f'(2.5)
last interval I chose to evaluate f'(4)

Answer 21

f'>0 means f is increasing
f'<0 means f is decreasing

Answer 22

ok...

Answer 23

so have you evaluate f'(0) and f'(2.5) and f'(4)

Answer 24

we are trying to confirm what you said what happens at x=e
you said there was a relative min at x=e
if that is so then the function will switch from decreasing to increasing there

Answer 25

ok? but i was taking a wild stab at x= e i am not sure if it is exactly a relative minimum...

Answer 26

right @freckles
and if the function switches from increasing to decreasing then it would be a relative max

Answer 27

so have you done what freckles as asked yet

Answer 28

evaluated f'(0), f'(2.5), and f'(4)?

Answer 29

this will tell us what is happening on those 3 intervals

Answer 30

im plugging it in but 0 is undefined what does that mean?

Answer 31

oh yeah we have ln(x) there
plug in a number like 1 instead

Answer 32

ok so f'(2.5) is around .04

Answer 33

so f'(2.5) is positive and f'(4) is negative

Answer 34

so we know its increasing, decreasing and then increasing?

Answer 35

so you got f'(1) is positive or negative?

Answer 36

if f'>0 then f is increasing
if f'<0 then f is decreasing

Answer 37

ok now now we know 3 is minimum?

Answer 38

is this what you got