Show that $\left(\dfrac{1}{(x\ln x(\ln(\ln x))^{2/3}}\right)$ is decreasing for all $x > 3$

Question

I am having trouble with this problem. I would rather not take the derivative of this function and show it like that. Is there any other way of showing it?

Answer 1

Well certainly, $(x\log x \log(\log(x)))^{2/3}$ is increasing for $x>3$. What does that tell you about the reciprocal?

Answer 2

$y^{2/3}$ is decreasing iff $y$ is decreasing, $\frac1y$ is decreasing iff $y$ is increasing (and never $=0$). For $x>3>3$, $\ln x$ is increasing and $>1$, so $\ln\ln x$ is increasing and $>0$. With $x$, $\ln x$, and $\ln\ln x$ positve and increasing,so is their product.