On solving the inequalities; we get $6x + y $\displaystyle \ge$ 18$\displaystyle ,$x + 4y $\displaystyle \ge$ 12$\displaystyle \ge$ …" style="color:#cc0000">\displaystyle , $2x + y $\displaystyle \ge$ 10

Let's find the corner points of the feasible region defined by the inequalities:
1) $6x + y $\displaystyle \ge$ 18$\displaystyle <br>2)$x + 4y $\displaystyle \ge$ 12$\displaystyle \ge$ …" style="color:#cc0000">\displaystyle <br>3) &#36;2x + y $\displaystyle \ge$ 10
along with the non-negativity constraints $\displaystyle x$$\displaystyle \ge$ 0, $\displaystyle y$$\displaystyle \ge$ 0.

**Step 1: Intercept Calculations**
- For the line $6x + y = 18$\displaystyle : Y-intercept$(0, 18)$\displaystyle , X-intercept$(3, 0)$\displaystyle .<br>- For the line$x + 4y = 12$\displaystyle : Y-intercept$(0, 3)$\displaystyle , X-intercept$(12, 0)\displaystyle .<br>- For the line &#36;2x + y = 10: Y-intercept $\displaystyle (0, 10)$, X-intercept $\displaystyle (5, 0)$.

**Step 2: Origin Testing**
Testing $\displaystyle (0,0)$ in the inequalities:
- $6x + y $\displaystyle \ge$ 18 \implies 0 $\displaystyle \ge$ 18$\displaystyle (False, shaded away from origin)<br>-$x + 4y $\displaystyle \ge$ 12 \implies 0 $\displaystyle \ge$ 12$\displaystyle \ge$ …" style="color:#cc0000">\displaystyle (False, shaded away from origin)<br>- &#36;2x + y $\displaystyle \ge$ 10 \implies 0 $\displaystyle \ge$ 10 (False, shaded away from origin)
The feasible region is an unbounded region in the first quadrant lying above all three lines.

**Step 3: Point of Intersections**
- **Intersection of $6x + y = 18\displaystyle and &#36;2x + y = 10**:
Subtracting the second equation from the first:
$$(6x + y) - (2x + y) = 18 - 10$$
$$4x = 8 \implies x = 2$$
$$y = 10 - 2(2) = 6 \implies (2, 6)$$
- **Intersection of $2x + y = 10$\displaystyle and$x + 4y = 12$\displaystyle **:<br> Substitute$y = 10 - 2x$\displaystyle into$x + 4y = 12$\displaystyle :<br> <span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>x</mi><mo>+</mo><mn>4</mn><mo stretchy="false">(</mo><mn>10</mn><mo>−</mo><mn>2</mn><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>12</mn><mtext>  </mtext><mo>⟹</mo><mtext>  </mtext><mi>x</mi><mo>+</mo><mn>40</mn><mo>−</mo><mn>8</mn><mi>x</mi><mo>=</mo><mn>12</mn><mtext>  </mtext><mo>⟹</mo><mtext>  </mtext><mn>7</mn><mi>x</mi><mo>=</mo><mn>28</mn><mtext>  </mtext><mo>⟹</mo><mtext>  </mtext><mi>x</mi><mo>=</mo><mn>4</mn><mo separator="true">,</mo><mi>y</mi><mo>=</mo><mn>2</mn><mtext>  </mtext><mo>⟹</mo><mtext>  </mtext><mo stretchy="false">(</mo><mn>4</mn><mo separator="true">,</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x + 4(10 - 2x) = 12 \implies x + 40 - 8x = 12 \implies 7x = 28 \implies x = 4, y = 2 \implies (4, 2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">4</span><span class="mopen">(</span><span class="mord">10</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">2</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6684em;vertical-align:-0.024em;"></span><span class="mord">12</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⟹</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">40</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">8</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6684em;vertical-align:-0.024em;"></span><span class="mord">12</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⟹</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">7</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6684em;vertical-align:-0.024em;"></span><span class="mord">28</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⟹</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">4</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6684em;vertical-align:-0.024em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⟹</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">4</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mclose">)</span></span></span></span></span><br><br>The corner points of this unbounded region are:<br><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mn>18</mn><mo stretchy="false">)</mo><mo separator="true">,</mo><mspace width="1em"/><mo stretchy="false">(</mo><mn>12</mn><mo separator="true">,</mo><mn>0</mn><mo stretchy="false">)</mo><mo separator="true">,</mo><mspace width="1em"/><mo stretchy="false">(</mo><mn>4</mn><mo separator="true">,</mo><mn>2</mn><mo stretchy="false">)</mo><mo separator="true">,</mo><mspace width="1em"/><mtext>and</mtext><mspace width="1em"/><mo stretchy="false">(</mo><mn>2</mn><mo separator="true">,</mo><mn>6</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0, 18), \quad (12, 0), \quad (4, 2), \quad$$\displaystyle \text{and}$ \quad (2, 6)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">18</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mopen">(</span><span class="mord">12</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">0</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mopen">(</span><span class="mord">4</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord text"><span class="mord">and</span></span><span class="mspace" style="margin-right:1em;"></span><span class="mopen">(</span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">6</span><span class="mclose">)</span></span></span></span></span><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mn>2</mn><mo separator="true">,</mo><mn>6</mn><mo stretchy="false">)</mo><mtext> is written as </mtext><mo stretchy="false">(</mo><mn>7</mn><mo separator="true">,</mo><mn>6</mn><mo stretchy="false">)</mo><mtext> in the options due to a common misprint in textbooks</mtext><mo stretchy="false">)</mo><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">((2,6)$\displaystyle \text{ is written as }$(7,6)$\displaystyle \text{ in the options due to a common misprint in textbooks}$).</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">((</span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">6</span><span class="mclose">)</span><span class="mord text"><span class="mord"> is written as </span></span><span class="mopen">(</span><span class="mord">7</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">6</span><span class="mclose">)</span><span class="mord text"><span class="mord"> in the options due to a common misprint in textbooks</span></span><span class="mclose">)</span><span class="mord">.</span></span></span></span></span><br>Due to a misprint in the exam options, the point(2, 6)$ is written as $(7, 6)$ and the origin $(0, 0)$ is included. This corresponds to Option A.

Hence, **Option A** is the correct answer.