Step-by-Step Solution

The problem requires us to compare the values of the objective function $Z = x + 2y$ at different points in the feasible region. Since the coordinates of P, Q, and R are not explicitly given, we will assume some coordinates for demonstration. Let's assume P = (1,1), Q = (2,2), and R = (3,1). These points are chosen arbitrarily to represent points within a feasible region. 1. Value of Z at P: $Z_P = 1 + 2(1) = 3$ 2. Value of Z at Q: $Z_Q = 2 + 2(2) = 6$ 3. Value of Z at R: $Z_R = 3 + 2(1) = 5$ Now, let's evaluate the given options: (A) Z is minimum at $(\frac{18}{7}, \frac{2}{7})$: We cannot verify this without the actual feasible region. (B) Z is maximum at R$(\frac{7}{2}, \frac{3}{4})$: We cannot verify this without the actual feasible region. (C) (Value of Z at P) > (Value of Z at Q): Based on our assumed points, $Z_P = 3$ and $Z_Q = 6$. So, $Z_P < Z_Q$. This statement is incorrect. (D) (Value of Z at Q) < (Value of Z at R): Based on our assumed points, $Z_Q = 6$ and $Z_R = 5$. So, $Z_Q > Z_R$. This statement is incorrect. However, without the exact coordinates of P, Q, and R from the graph, we cannot definitively determine the correct answer. We need to analyze the graph to find the coordinates of P, Q, and R. Let's assume from the graph that P = (1, 2), Q = (2, 4), and R = (4, 1). 1. Value of Z at P: $Z_P = 1 + 2(2) = 5$ 2. Value of Z at Q: $Z_Q = 2 + 2(4) = 10$ 3. Value of Z at R: $Z_R = 4 + 2(1) = 6$ (C) (Value of Z at P) > (Value of Z at Q): $5 > 10$ is false. (D) (Value of Z at Q) < (Value of Z at R): $10 < 6$ is false. Let's consider another set of points based on a possible feasible region: P(0, 0), Q(1, 2), R(2, 1). Z at P = 0 + 2(0) = 0 Z at Q = 1 + 2(2) = 5 Z at R = 2 + 2(1) = 4 (C) Z(P) > Z(Q) => 0 > 5 False (D) Z(Q) < Z(R) => 5 < 4 False Without the graph, it's impossible to give a definitive answer. However, the approach is to find the coordinates of P, Q, and R from the graph and then evaluate the objective function at these points to compare the values. Since I cannot see the graph, I will make an educated guess based on the general principles of LPP. In a bounded feasible region, the maximum and minimum values occur at the corner points. Options (A) and (B) suggest specific points for minimum and maximum, which might be correct. Options (C) and (D) compare the values at different points. Let's assume that after analyzing the graph, we find that Q has a higher Z value than R. Then option (D) would be incorrect. If we find that Z is maximum at R, then option (B) would be correct. Given the options, let's assume option (D) is correct: (Value of Z at Q) < (Value of Z at R).