I've been trying to find (analytically) the maximum of the function
$$ f(x,\lambda,t) = \frac{1}{2}\left(1-\frac{x}{\lambda} \right)\left[\sqrt{\left(t+\frac{x^2}{1+x^2}\right)^2 + \frac{x^2}{(1+x^2)^2} } + \left(t+\frac{x^2}{1+x^2}\right)\right],$$
with respect to $x$, in the interval $0<x<\lambda$. Here, $\lambda$ and $t$ can simply be treated as constants (with, generally, $t\sim 1$). Setting the first derivative to zero yields
$$ 0 = \frac{x}{(1+x^2)^2}(\lambda -x)\left[ 2\sqrt{\left(t+\frac{x^2}{1+x^2}\right)^2 + \frac{x^2}{(1+x^2)^2} } + (2t+1) \right]- \sqrt{\left(t+\frac{x^2}{1+x^2}\right)^2 + \frac{x^2}{(1+x^2)^2} }\left[\sqrt{\left(t+\frac{x^2}{1+x^2}\right)^2 + \frac{x^2}{(1+x^2)^2} } + \left(t+\frac{x^2}{1+x^2}\right)\right], $$
which I cannot seem to solve analytically; even my computer finds this difficult.
Does anyone have any other techniques I could try in order to determine the location of this maximum in terms of $\lambda$ and $t$? Even constraining the location of the maximum will be useful, but I am unsure of how to do this!