Let $\Omega$ be a non-empty bounded open set in $\mathbb{R}^{2}$ with closure $\bar{\Omega}$ and boundary $\partial \Omega$ and let $\phi: \bar{\Omega} \rightarrow \mathbb{R}$ be a continuous function. Give a proof or a counterexample for each of the following assertions.

(i) If $\phi$ is twice differentiable on $\Omega$ with $\nabla^{2} \phi(\mathbf{x})>0$ for all $\mathbf{x} \in \Omega$, then there exists an $\mathbf{x}_{0} \in \partial \Omega$ with $\phi\left(\mathbf{x}_{0}\right) \geqslant \phi(\mathbf{x})$ for all $\mathbf{x} \in \bar{\Omega}$.

(ii) If $\phi$ is twice differentiable on $\Omega$ with $\nabla^{2} \phi(\mathbf{x})<0$ for all $\mathbf{x} \in \Omega$, then there exists an $\mathbf{x}_{0} \in \partial \Omega$ with $\phi\left(\mathbf{x}_{0}\right) \geqslant \phi(\mathbf{x})$ for all $\mathbf{x} \in \bar{\Omega}$.

(iii) If $\phi$ is four times differentiable on $\Omega$ with

$$\frac{\partial^{4} \phi}{\partial x^{4}}(\mathbf{x})+\frac{\partial^{4} \phi}{\partial y^{4}}(\mathbf{x})>0$$

for all $\mathbf{x} \in \Omega$, then there exists an $\mathbf{x}_{0} \in \partial \Omega$ with $\phi\left(\mathbf{x}_{0}\right) \geqslant \phi(\mathbf{x})$ for all $\mathbf{x} \in \bar{\Omega}$.

(iv) If $\phi$ is twice differentiable on $\Omega$ with $\nabla^{2} \phi(\mathbf{x})=0$ for all $\mathbf{x} \in \Omega$, then there exists an $\mathbf{x}_{0} \in \partial \Omega$ with $\phi\left(\mathbf{x}_{0}\right) \geqslant \phi(\mathbf{x})$ for all $\mathbf{x} \in \bar{\Omega}$.