Question

PLEASE Calculus Help?

(a) find the critical numbers of f (if any), (b) find the open interval(s) on which the function is increasing or decreasing, (c) apply the First Derivative Test to identify all relative extrema, and (d) use a graphing utility to confirm your
results .

2. f(x)= x/x+1

Answer 1

I already know I have to start by finding the derivative which is f'(x) = 1/ (x+1)^2 right?

Answer 2

and the roots would be x=-1

Answer 3

?

Answer 4

that derivative doesnt look right

Answer 5

I did it wrong'?

Answer 6

can you help me?

Answer 7

for the derivative you can use the quotient rule

f'(u/v) = (u'v -v'u)/v^2 where u=x and v=x+1

Answer 8

the critical numbers would be x=0 and x=-1

because f(0) = 0 and f(-1) is undefined

Answer 9

ok

Answer 10

also the lim x->inf = 1 and lim x->-inf = 1

Answer 11

so would the derivative be x(x+1) - x+1(x)/(x+1)^2 ?

Answer 12

is that the answer for b?

Answer 13

should be x+1 not x=1

Answer 14

x' = 1 and (x+1)' = 1

do you follow?

Answer 15

I don't understand... is that part b? or are you talking about the derivative?

Answer 16

starting again

it should be [x'(x+1) - (x+1)'x]/(x+1)^2

Answer 17

for supporting only, @alekos http://www.wolframalpha.com/input/?i=%28x%2F%28x%2B1%29%29%27

Answer 18

ok

Answer 19

... I'm confused now.

Answer 20

your derivative is perfectly OK. f'(x) = \(\dfrac{1}{(1+x)^2}\)

Answer 21

and it never =0, it's undefined when denominator =0, it means x =-1, so far so good?

Answer 22

yes.

Answer 23

my apologies you are right

it comes to 1/(x+1)^2

Answer 24

ok, go ahead @alekos

Answer 25

and the critical numbers are correct yes?

Answer 26

yes they are

Answer 27

it's another case to study, friend.
1 > 0 always, OK?
now denominator, except x = -1, x +1 =0
for all values of x, (x+1)^2 >0
got this part?

Answer 28

for example, if x =-2, x +1 = -1, then (x+1)^2 = (-1)^2 =1 >0
if x = 5 , x +1 =6 then (x+1)^2 = 6^2 =36>0

Answer 29

right. I got x=-1 and alekos showed me x=0. I understand that

Answer 30

yes I get that

Answer 31

ok, now make the table