Question

For any constant a, let
g(x) = ax − 2xln(x)
for
x > 0.


(a) What is the x-intercept
g(x)?

x =

Answer 1

solve
\[ax+2x\ln(x)=0\] i guess

Answer 2

\[2x\ln(x)=-ax\] is a start
then since you know \(x\neq 0\) i think you can go ahead and divide both sides by \(x\) and get \[2\ln(x)=-a\] or
\[\ln(x)=-\frac{a}{2}\]

Answer 3

hmm maybe that is wrong maybe factoring is better
\[x(a+2\ln(x))=0\] nope works the same way

Answer 4

yeah I had that as my original answer and got it wrong ;/ I'm lost on how to do it a different way

Answer 5

Im also trying to find the x coordinate of the critical point of g(x)

Answer 6

to find the x-intercept set y=0 and solve for x

Answer 7

\[ax-2x \ln(x)=0\\ x(a-2 \ln(x))=0 \\ x \neq 0 \\ a-2 \ln(x)=0 \\ \text{ finish solving for x }\]

Answer 8

\[2 \ln(x)=a \]

Answer 9

so e^a/2?

Answer 10

yeah
(e^(a/2), 0) <--x-intercept

Answer 11

okay thanks! so to find the x vale of the critical point of g(x) i just set the first derivative to zero and solve for x..?

Answer 12

solve g'=0
g' dne
and make sure those numbers occur in the domain
those are critical numbers

Answer 13

ahhh i see! Okay thank you! I was able to solve it!

Answer 14

coolness