Question

Find the -coordinate of the point where the graph of the function

(ln(sqrtx))/x^2

has a horizontal tangent line.

what is the x-coordinate

Answer 1

\[\ln (\sqrt{x})/x^2\]

Answer 2

first off
\[\frac{\ln(\sqrt{x})}{x^2}=\frac{1}{2}\frac{\ln(x)}{x^2}\] now we need the derivative

Answer 3

right

Answer 4

so im not sure where to go from here.

Answer 5

\[\frac{1-2\ln(x)}{2x^3}\] is the derivative

Answer 6

set numerator = 0 and solve for x

Answer 7

wait how did you get 1-2ln(x)/2x^3

Answer 8

\[2\ln(x)=1\]
\[\ln(x)=\frac{1}{2}\]
\[x=e^{\frac{1}{2}}=\sqrt{e}\]

Answer 9

quotient rule

Answer 10

use
\[( \frac{f}{g})'=\frac{gf'-fg'}{g^2}\] with
\[f(x)=\ln(x), f'(x)=\frac{1}{x}, g(x)=x^2,g'(x)=2x\]

Answer 11

using 1/2 as a function and ln(x)/x^2 as another function? and why wouldn't that be product rule

Answer 12

you have a quotient, not a produce
the 1/2 is a constant, so just leave it there

Answer 13

ok that makes sense i see what your doing

Answer 14

*product

Answer 15

so the final answer is e^1/2?

Answer 16

that is what i got, yes

Answer 17

ok sounds good to me