Let $X_1$ and $X_2$ be random variables (not of the same distribution and not independent). Both have a zero probability of being below $-1$. Their joint density is $\rho(x_1,x_2)$. Also, they both have finite expectations.
Now, define the region $A = \{ (t_1,t_2)\in\mathbb{R}^2 \mid 0\le t_1,t_2<1 \text{ and }t_1+t_2<1 \}$, and define the function $f:A\to\mathbb{R}$ with
$f(t_1,t_2) = \int_{-1}^\infty\int_{-1}^\infty \log(1+t_1x_1+t_2x_2)\rho(x_1,x_2)\,dx_1dx_2.$
Can we say something interesting about $f$? For example, does $f$ have at most only one local maximum?