Is $\log(x)|1-x/n|^n$ bounded?

Question

I have to find if the function $\log(x)|1-x/n|^n$ is bounded but I cannot. I have tried Bernoulli's inequality and a few others things. Any hints please?

Thanks.

Answer 1

The function $f(x) = log(x)|1-x/n|^n$ is unbounded as a function of $x$.

If $x > e$, $f(x) > |1-x/n|^n$.

Further, if $x > n$, $f(x) > (\frac{x-n}{n})^n$ which clearly diverges as $x$ gets large.

On the lower end, as $x\rightarrow 0$, $log(x)\rightarrow -\infty$ while the absolute value term goes to 1.