Can anyone help me to prove the convexity/concavity of following complicated function...? I have tried a lot of methods (definition, 1st derivative etc.), but this function is so complicated, and I finally couldn't prove... however, I plotted with many different parameters, it's always concave to $\rho$...
$ f\left( \rho \right) = \frac{1}{\lambda }(M\lambda \phi - \rho (\phi - \Phi )\ln (\rho + M\lambda ) + \frac{1}{{{e^{(\rho + M\lambda )t}}\rho + M\lambda }}\cdot( - (\rho + M\lambda )({e^{(\rho + M\lambda )t}}{\rho ^2}t(\phi - \Phi ) ) $ $+ M\lambda (\phi + \rho t\phi - \rho t\Phi )) + \rho ({e^{(\rho + M\lambda )t}}\rho + M\rho )(\phi - \Phi )\ln ({e^{(\rho + M\lambda )t}}\rho + M\lambda ))$
Note that $\rho > 0$ is the variable, and $M>0, \lambda>0, t>0, \phi>0, \Phi>0 $ are constants with any possible positive values...