Fixed-charge Reissner-Nordstrom temperature maximum ($M_{\max }=2|Q|/\sqrt{3},T_{\max }=1/(6\sqrt{3}\pi |Q|)$)

Set $y=r_{+}/|Q|\ge 1$. Then $M/|Q|=(y+y^{-1})/2$ and $|Q|T=(y^{-1}-y^{-3})/(4\pi )$. Differentiation gives a unique maximum at $y=\sqrt{3}$, proving the displayed mass and temperature. The curve is zero at extremality and asymptotic to $1/(8\pi M)$ at large mass.