Where are the inflection points, if any exist, for $f(x)=10\frac{\ln(\ln(x))}{\ln(x)}$?
I don't think any exist. But could use a little help. My guess would be that with the second derivative, if $x = e^{-2}$ you get 0, but this is not in the domain of the function. Is this correct, or completely off? Any help would be appreciated.
[Math] Inflection points with natural logs.
calculusderivatives
Best Answer
Inflection point must exist.
Indeed we have a maximum at $x=e^e$ and $f(x)\to 0$ as $x\to +\infty$, therefore concavity must change because $f(x)$ has no intersection with $x$-axis after $x=e$.
We have $$f''(x)=10\,\frac{ (\log x+2) \log (\log x)-\log x-3}{x^2 \log ^3x}$$
$f''(x)=0$ if $$(\log x+2) \log (\log x)-\log x-3=0$$ substitute $\log x=u$ $$(u+2)\log u-u-3=0\to u=3.28455$$ Finally we have $x=e^{3.28455}\approx 26.7$
the graph below shows the curve $y=f(x)$, the inflection point and the tangent
$$...$$