Suppose that $A x \leqslant b$ and $x \geqslant 0$ and $A^{T} y \geqslant c$ and $y \geqslant 0$ where $x$ and $c$ are $n$-dimensional column vectors, $y$ and $b$ are $m$-dimensional column vectors, and $A$ is an $m \times n$ matrix. Here, the vector inequalities are interpreted component-wise.

(i) Show that $c^{T} x \leqslant b^{T} y$.

(ii) Find the maximum value of

$$\begin{aligned} 6 x_{1}+8 x_{2}+3 x_{3} \quad \text { subject to } & 2 x_{1}+4 x_{2}+x_{3} \leqslant 10 \\ & 3 x_{1}+4 x_{2}+3 x_{3} \leqslant 6 \\ & x_{1}, x_{2}, x_{3} \geqslant 0 \end{aligned}$$

You should state any results from the course used in your solution.