Minimum of the sum of two functions

Question

I want to show that trying to find the minimum of the sum of two or more functions of two different groups is a not convex problem. For example: $ \min\limits_{Y,Z} f(X,Y,Z)=...$. Moreover the values $X,Y,Z$ are matrices. My idea is to show that the Hesse-Matrix of the sum of those added functions is not always positive semidefinit by finding a point x, where it is not psd?
$\operatorname{H}_f({Y},{Z})= \left(\frac{\partial^2f}{\partial c_i\partial c_j}({Y},{Z})\right) \begin{pmatrix} \frac{\partial^2 f}{\partial c_1\partial c_1}({X},{Y})&\frac{\partial^2 f}{\partial c_1\partial c_2}({X},{Y})&\cdots&\frac{\partial^2 f}{\partial c_1\partial c_n}({X},{Y})\\[0.5em] \frac{\partial^2 f}{\partial c_2\partial c_1}({X},{Y})&\frac{\partial^2 f}{\partial c_2\partial c_2}({X},{Y})&\cdots&\frac{\partial^2 f}{\partial c_2\partial c_n}({X},{Y})\\ \vdots&\vdots&\ddots&\vdots\\ \frac{\partial^2 f}{\partial c_n\partial c_1}({X},{Y})&\frac{\partial^2 f}{\partial c_n\partial c_2}({X},{Y})&\cdots&\frac{\partial^2 f}{\partial c_n\partial c_n}({X},{Y}) \end{pmatrix}$
with $c_i \in X,Y$, for example $c_i=x_{11}$ Then I pick one $X,Y,Z$ and show the the eigenvalues are $<0$. P.S. log is elementwise