Question

Find the critical numbers of the function. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)
f(x) = x^−6 ln x

Answer 1

the actual question is \[x^{-6}\ln x\]
I think the derivative is \[x^{-7}-6x^{-7}\ln x \]

Answer 2

derivative is better written as
\[\frac{1-6\ln(x)}{x^7}\] makes finding the critical points a lot easier

Answer 3

on critical point is 0, but that is not in the domain of your function. so you can ignore it
other one solve
\[1-6\ln(x)=0\]

Answer 4

i think part of my problem is i don't know how to solve for ln

Answer 5

\[1-6\ln(x)=0\]
\[6\ln(x)=1\]
\[\ln(x)=\frac{1}{6}\]
\[x=e^{\frac{1}{6}}\]

Answer 6

like that
you are going to have to "exponentiate" at some point

Answer 7

ok thanks, i know eulers number is related to natural log i just don't know how to convert back and forth. thanks for the help

Answer 8

\[\ln(x)=y\iff e^y=x\]

Answer 9

so if i had ln23 that would be e^23?

Answer 10

actually ln23 is 3.13 so E^3.13 = 23?