Q:

Finding max value

Which approach would be the best to take in order to calculate the max value of $a$, where $a\ \log(x+1)$ (blue) at no point exceeds $\sqrt{x}$ (red)?

A:

We want to find the maximum value of $\frac {\log (x+1)}{\sqrt x}$ so take the derivative and set to zero.  You get an expression that cannot (without the Lambert W function) be solved analytically.  Alpha gets about $x=3.92155$ where $\frac {\log(x+1)}{\sqrt x}\approx 0.804742$  The maximum $a$ is the inverse of this, about $1.24263$

