Proving a function is monotonic decreasing

Question

I need to prove that following function is monotonic decreasing for all $x > 1$ - $$F(x) = \frac{\sqrt{x}}{2^{x\sqrt{x}}}$$

So by getting the derivate is one way but in this case its a little bit more complicated - is there a simple way to recognize faster when a function is monotonic?

Answer 1

Let $\sqrt{x} = u. $ Then you get that $F(u) = \frac{u}{2^{u^3}}$.

$F'(u) = 2^{-u^3}(1 - u^3\log(8))$. For all $u>1 \Leftrightarrow \sqrt{x} > 1 \Leftrightarrow x>1$, your function is decreasing monotonically.