Question

relative extrema for y=x ln x

Answer 1

did you find the derivative?

Answer 2

need the product rule to find it
you gotta find that first

Answer 3

yes is lnx+1

Answer 4

so it is
then set it equal to zero, solve for \(x\)

Answer 5

i don't know how to solve it

Answer 6

too many typos
\[\ln(x)+1=0\\
\ln(x)=-1\] any idea what comes next?

Answer 7

hint, rewrite in equivalent exponential form

Answer 8

plug it in??

Answer 9

oh no

Answer 10

\[\ln(a)=b\iff a=e^b\]

Answer 11

oh so is x=e^-1

Answer 12

so
\[\ln(x)=-1\iff x=e^{-1}=\frac{1}{e}\]

Answer 13

right

Answer 14

now i guess you need to know whether that is a relative max, min, or neither

Answer 15

yea i have to identify that and also find the point of inflection

Answer 16

to identify it you have a choice, but probably the easiest method is to take the second derivative, especially since you have to find it anyways

Answer 17

what is the derivative of
\[\ln(x)+1\]?

Answer 18

1/(x+1)

Answer 19

Find the derivative separately.. :)

Answer 20

on no dear, that is the derivative of \(\ln(x+1)\)

Answer 21

the derivative of \(\ln(x)\)is \(\frac{1}{x}\) and the derivative of \(1\) is zero

Answer 22

i thought i need to find the derivative of ln (x+1)

Answer 23

\[y = x \cdot \ln(x) \\ y^{\prime} = \ln(x) + 1 \\ y^{''} = \frac{1}{x}\]

Answer 24

if you evaluate \(\frac{1}{x}\) at \(\frac{1}{e}\) you get \(e\) which is evidently positive
that means
a) the function is concave up and
b) the critical point gives a relative minimum

Answer 25

oh i get it

Answer 26

and for the point of inflection, i know you have to find the second derivative but i don't know what next

Answer 27

set it equal to zero and solve if you are not thinking straight

Answer 28

if you are thinking straight, then note that the domain of \[f(x)=x\ln(x)\] is \((0,\infty)\)
on that interval \(\frac{1}{x}>0\) for any positive \(x\)

Answer 29

meaning the function is always concave up, and there are no inflection points

Answer 30

alright i guess i know how to do it

Answer 31

thank you for your help~

Answer 32

yw