# Solving a particularly difficult exponential equation

I was recently looking for the maximum value of $x^{10-x}$ over $x>0$ without using calculus. I first thought of expanding it to $\frac{x^{10}}{x^x}$ and then looking for when $x^x$ grows faster or slower than $x^{10}$ by incrementing $x$ by 1 (I know, not very rigorous). From this, I could deduce over which intervals the function was increasing or decreasing, and from this, the maximum value. After some math, I arrived at the inequalities $$x^{10-x}>{(x+1)}^{9-x}$$ and $$x^{10-x}<{(x+1)}^{9-x}$$ This is where I got stuck. Is there any way to solve these two inequalities algebraically? The only option I could think of is finding an approximation through a graph, but of course an exact answer would always be more useful. Thanks!

Since $x^{9-x}>0$, $f'(x)=0$ when $10-x-x\ln x=0$, and this equation has one real solution in terms of Lambert W function as follows: