Final answer:
To minimize P=2x+y under the given constraints, a graphical approach shows that the minimum occurs when x=0 and y=0 within the feasible region. Since increasing x or y will increase P, the minimum value for P is thus 0.
Step-by-step explanation:
The objective is to minimize P = 2x + y under the constraints that x ≥ 0, y ≥ 0, and x + y ≤ 6. To minimize P given these constraints, we must find the lowest value for P while staying within the feasible region defined by the constraints. This problem is a linear programming problem and can be solved graphically or by using simplex method, but the graphical method is suitable for this context with two variables.
To solve it graphically, we plot the region that satisfies the constraints, known as the feasible region. Then, we plot several lines that represent different values of P. We look for the smallest value of P, where the corresponding line is still touching the feasible region. As the constraints suggest a triangular feasible region in the first quadrant, one of the corners will represent the minimum value for P. Since P = 2x + y, if either x or y is 0, P is minimized because we strictly increase the value of P by increasing x or y.
Based on the given constraints, the minimum value of P occurs at x = 0 and y = 0, in which case P = 0, because adding anything to x or y would increase the value of P. Therefore, the minimum value for P is 0, and both x and y are at their lower bounds.