Maximizing the sum of nonnegative functions.

Question

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!

Answer 1

Take $f(x)=(2-x)$ and $g(x)=\sqrt x$ for $x \in [0,2]$

Take $p=2$

$h_1(x)=f+g$

$h_2(x)=f^2+g^2$

$h_1$ is maximum for $x=\dfrac{1}{4}$ and $h_2$ is maximum for $x=0$ (edited after comment...), with its minimum for $\dfrac{3}{2}$