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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 $(x,u)$.

Anna D.
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    Just fix $t, p$ and consider the function $$g(x, u)=f(t,x,u,p).$$ You want to prove that $g$ is concave on $\mathbb R^2$. Then, study its second derivatives etc etc... This amounts to considering only the derivatives in $x, u$, as you conjecture. – Giuseppe Negro Dec 16 '18 at 17:02
  • @GiuseppeNegro Thank you, that is precisely that I wanted to know (whether you could do that). – Anna D. Dec 16 '18 at 17:07

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