Let a general multivariate function of $n$ variables, $f : \mathbb{R}^n \to \mathbb{R}$ say, be given.
Suppose we want to prove that $f$ is convex (concave) in just some of the $n$ variables, not all.
In general, if one wants to prove that a function is convex (concave) in all variables, one should use the Hessian of the function.
So, I suspect that if one wants to prove that a function is convex (concave) in only some of the variables, one should again use a matrix of second-order partial derivatives of the function but in this case ONLY with respect to the variables for one wishes to confirm convexity (concavity).
Is my supposition correct?
In my case I am dealing with a function $f : \mathbb{R}^4 \to \mathbb{R}, f(t,x,u,p) = 1 + x - u^2 + p(x + u)$ where I (only) want to show that it is concave in $(x,u)$.