Quotient Rule Question

[jtw2e2]

13). f(x) = ln x / x[sup:25ayje03]1/2[/sup:25ayje03]

a). Find the intervals on which f is increasing or decreasing
b). Find the local maximum and minimum values of f
c). Find the intervals of concavity and the inflection points

I'm actually on part c and trying to find f''(x).
So:
f(x) = ln x / x[sup:25ayje03]1/2[/sup:25ayje03]
f'(x) = (2 - ln x) / 2x[sup:25ayje03]3/2[/sup:25ayje03]

f''(x) =

I know this is idiotic, but I can't simplify that 2-lnx * 3x^(1/2). One of the biggest things messing me up is the minus out in front. I would assume that I should distribute it but the poorly-named "Solutions Manual" does not.

Please help. Thank you.

 

[daon]

The denominator is always positive for the domain of the numerator, hence you just need to solve numerator <0, numerator > 0

 

[BigGlenntheHeavy]

\(\displaystyle f(x) \ = \ \frac{ln|x|}{\sqrt x} \ f'(x) \ = \ \frac{2-ln|x|}{2x^{3/2}}, \ \implies \ ln|x| \ = \ 2 \ \implies \ x \ = \ e^{2}\)

\(\displaystyle Hence, \ f(e^{2}) \ = \ 2/e \ is \ a \ max.\)

\(\displaystyle Point \ of \ Inflection, \ x \ = \ e^{8/3}\)

\(\displaystyle f"(x) \ = \ \frac{3ln|x|-8}{4x^{5/2}}, \ \implies \ 3ln|x| \ = \ 8 \ \implies \ x \ = \ e^{8/3}, \ see \ graph.\)

[attachment=0:284yamj3]ghi.jpg[/attachment:284yamj3]