For any $r>0$ and positive integer $n$ define the following function
$$f_r(n) = \frac{1}{(nr)^{1/5}} – 4 \log \Big(\frac{3}{n^2}\Big)$$
I plotted the above function for multiple values of $r$ and it seems to be always greater than zero for any $n \geq 2$. How can I prove this?
At first, I thought maybe the derivative with respect to $n$ is always positive (and thus the function is positive for $n \geq c$, where $f_r(c) = 0$) but this turned out not to be the case:
$$\frac{\partial f_r(n)}{\partial n} = -\frac{1}{5} r^{-1/5}n^{-6/5} + 8n^{-1},$$
which can be negative when $r$ is very small.
Best Answer
$\log(3/n^2) < 0$ for $n \ge 2$, so $f_r(n)$ is the sum of two positive numbers.