I was trying to solve the problem A maximization problem when I ask myself if the general problem
\begin{equation} \begin{array}{c} maximize\hspace{1cm} f(\mathbf{X})^p +g(\mathbf{X})^p \\ s.t. \hspace{1cm} \mathbf{X} \in K \subseteq \mathbb{R}^{m \times n}, \end{array} \end{equation} is equivalent to
\begin{equation} \begin{array}{c} maximize\hspace{1cm} f(\mathbf{X}) +g(\mathbf{X}) \\ s.t. \hspace{1cm} \mathbf{X} \in K \subseteq \mathbb{R}^{m \times n}, \end{array} \end{equation} when the scalar functions $f(\mathbf{X})$ and $g(\mathbf{X})$ are nonnegative on $K$, and $p > 0$.
Is this true? If not, how to find a counterexample?
Thanks in advance!