Question

If corner points of a feasible region are (0, 0), (2,0) \((\frac{20}{19},\frac{45}{19})\) and (0, 3), then
(A) Maximum value of z=5x+3y is 10
(B) Minimum value of z=5x+3y is 0
(C) Maximum value of z=5x+3y is \(\frac{235}{19}\) and minimum value is 0
(D) Maximum value of z=5x+3y is 10 and minimum value is 0
Choose the correct answer from the options given below :

• A. (A) and (D) Only
• B. (B) and (D) Only
• C. (B) and (C) Only
• D. (A), (B) and (D) Only

Answer 1

The Correct Option is C

To solve for the maximum and minimum values of \(z = 5x + 3y\) using the given corner points of the feasible region, we evaluate \(z\) at each vertex:

• At (0, 0): \(z = 5(0) + 3(0) = 0\)
• At (2, 0): \(z = 5(2) + 3(0) = 10\)
• At \(\left(\frac{20}{19}, \frac{45}{19}\right):\) \(z = 5\left(\frac{20}{19}\right) + 3\left(\frac{45}{19}\right) = \frac{100}{19} + \frac{135}{19} = \frac{235}{19}\)
• At (0, 3): \(z = 5(0) + 3(3) = 9\)

From these calculations, the minimum value of \(z\) is 0 at the point (0, 0), and the maximum value of \(z\) is \(\frac{235}{19}\) at the point \(\left(\frac{20}{19}, \frac{45}{19}\right)\).

The correct options reflecting these values are (B) and (C) only.