# Show that maximisers are in corners

Let $\Pi$ be a rectangle $[a,b] \times [c,d]$ containing $0$: $a < 0 < b$, $c < 0 < d$, let $f(x)$ be a convex continuous function on $\Pi$. Define a functional $$J(x_1,x_2,x_3) = p_1 f(x_1) + p_2 f(x_2) + p_3 f(x_3),$$ where $p_1,p_2,p_3$ are given by system $$p_1 + p_2 + p_3 = 1, \\ p_1 x_1^1 + p_2 x_2^1 + p_3 x_3^1 = 0, \\ p_1 x_1^2 + p_2 x_2^2 + p_3 x_3^2 = 0.$$ When determinant of this system is not equal to zero we have $$J(x_1,x_2,x_3) = \frac{\begin{vmatrix} f(x_1) & f(x_2) & f(x_3) \\ x_1^1 & x_2^1 & x_3^1 \\ x_1^2 & x_2^2 & x_3^2 \end{vmatrix}}{\begin{vmatrix} 1 & 1 & 1 \\ x_1^1 & x_2^1 & x_3^1 \\ x_1^2 & x_2^2 & x_3^2 \end{vmatrix}}$$ How to show that maximum of $J(x_1,x_2,x_3)$ over $x_1,x_2,x_3 \in \Pi$ is given by $x_1,x_2,x_3$ in corners of $\Pi$?

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