let $f(x)=xa^{\frac{1}{x}}-x$ for positive $x,a\in \mathbb{R}$
Show that $f(x)$ is decreasing on $(0, \infty)$.
I am trying to show that $f'(x)= a^{\frac{1}{x}}(1-\frac{ln(a)}{x})-1$ is always negative.
To do that, I am thinking about showing that finding a max/min for $f'(x)$ by solving for $f''(x) = 0$ then prove that what we found was indeed a maximum by calculating $f'''(x)$ but then things get ugly.
So I'm wondering if there is a easier way.
