Hi, I'm trying to prove that $f(x) = \frac{e^x}{x^x}$ is decreasing on interval $[1, \infty)$.

Know that the statement I want to prove is as follows: $\forall a, b \in [1, \infty), a < b \Rightarrow \frac{e^a}{a^a} \geq \frac{e^b}{b^b}$.

I start off with a < b, then try to reach $\frac{e^a}{a^a} \geq \frac{e^b}{b^b}$ through some algebraic manipulations.

$$a < b$$

$$a^a < b ^b$$

$$\frac{1}{a^a} > \frac{1}{b^b}$$

$$\frac{e^a}{a^a} > \frac{e^a}{b^b}$$

Then what?