Given constraints: $$ x \geq 0, y \geq 0,2 x+2 y \geq 4, x+y \leq 8 $$ Explain the steps for maximizing the objective function $$ P=3 x+4 y $$.

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**Step 1. Write Down the Constraints and Objective Function**

We are given the constraints:
$$
x \geq 0,\quad y \geq 0,\quad 2x+2y \geq 4,\quad x+y \leq 8
$$
and the objective function to maximize is
$$
P = 3x + 4y.
$$

**Step 2. Simplify the Constraints**

Divide the inequality $$2x+2y \geq 4$$ by 2 to obtain:
$$
x+y \geq 2.
$$
Thus, the constraints become:
$$
x \geq 0,\quad y \geq 0,\quad x+y \geq 2,\quad x+y \leq 8.
$$

**Step 3. Identify the Feasible Region**

The constraints $$x \geq 0$$ and $$y \geq 0$$ restrict the solution to the first quadrant. The inequalities
$$
x+y \geq 2 \quad \text{and} \quad x+y \leq 8
$$
describe two parallel lines. The feasible region is the set of points in the first quadrant that lie between (or on) the lines:
$$
x+y = 2 \quad \text{and} \quad x+y = 8.
$$

**Step 4. Determine the Corner Points (Vertices) of the Feasible Region**

Find the intersections of the constraint boundaries:

1. Intersection of $$x=0$$ and $$x+y=2$$:
   $$
   0+y=2 \quad \Rightarrow \quad y=2 \quad \Rightarrow \quad (0,2).
   $$

2. Intersection of $$y=0$$ and $$x+y=2$$:
   $$
   x+0=2 \quad \Rightarrow \quad x=2 \quad \Rightarrow \quad (2,0).
   $$

3. Intersection of $$x=0$$ and $$x+y=8$$:
   $$
   0+y=8 \quad \Rightarrow \quad y=8 \quad \Rightarrow \quad (0,8).
   $$

4. Intersection of $$y=0$$ and $$x+y=8$$:
   $$
   x+0=8 \quad \Rightarrow \quad x=8 \quad \Rightarrow \quad (8,0).
   $$

Thus, the vertices of the feasible region are:
$$
(0,2),\quad (2,0),\quad (0,8),\quad (8,0).
$$

**Step 5. Evaluate the Objective Function at Each Vertex**

Substitute each vertex into the objective function $$P = 3x + 4y$$:

1. At $$(0,2)$$:
   $$
   P = 3(0) + 4(2) = 0 + 8 = 8.
   $$

2. At $$(2,0)$$:
   $$
   P = 3(2) + 4(0) = 6 + 0 = 6.
   $$

3. At $$(0,8)$$:
   $$
   P = 3(0) + 4(8) = 0 + 32 = 32.
   $$

4. At $$(8,0)$$:
   $$
   P = 3(8) + 4(0) = 24 + 0 = 24.
   $$

**Step 6. Determine the Maximum Value**

Among the calculated values, the maximum value is:
$$
P = 32 \quad \text{at the point} \quad (0,8).
$$

**Conclusion**

The objective function $$P = 3x + 4y$$ is maximized at $$(x,y) = (0,8)$$ with a maximum value of
$$
32.
$$
To maximize $$ P = 3x + 4y $$ with the constraints $$ x \geq 0 $$, $$ y \geq 0 $$, $$ x + y \geq 2 $$, and $$ x + y \leq 8 $$, evaluate $$ P $$ at the vertices $$(0,2)$$, $$(2,0)$$, $$(0,8)$$, and $$(8,0)$$. The maximum value of $$ P $$ is 32 at $$(0,8)$$.

Given constraints:

Explain the steps for maximizing the objective
function .

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