I have a function that is of the form
$C({\bf x}) = c_1\left(a_1x_1 + b_1x_1^2\right) + c_2\left(a_2(x_1-x_2) + b_2(x_1-x_2)^2\right) + c_3\left(a_3(x_2-x_3) + b_3(x_2-x_3)^2\right)$
where each $c_i:\mathbb{R}\to\mathbb{R}$ is strictly convex and strictly increasing and all $a_i,b_i>0$. I want to determine if $C({\bf x})$ is strictly convex in ${\bf x} = (x_1,x_2,x_3)$. Is $c_1\left(a_1x_1 + b_1x_1^2\right)$ strictly convex in ${\bf x}$?
No, $c_1(a_1 x_1+ b_1 x_1^2)$ is not strictly convex in $\bf x$, because it is constant in $x_2$ and $x_3$.
But (and this is a hint) $c_2\left(a_2(x_1 - x_2) + b_2 (x_1 - x_2)^2\right)$ is strictly convex in $x_2$ and $c_3\left(a_3 (x_2 - x_3) + b_3 (x_2 - x_3)^2\right)$ is strictly convex in $x_3$.