The maximum value of Z=5x+4y, Subject to y≤2x,x≤2y,x+y≤3,x≥0,y≥0 is ..

MHT CET · Maths · Area Under Curves

The maximum value of $Z = 5 x + 4 y,$ Subject to $y \leq 2 x, x \leq 2 y, x + y \leq 3, x \geq 0, y \geq 0$ is ..

A 14
B 12
C 13
D 18

Verified Solution

Answer & Solution

Correct Answer

(A) 14

Step-by-step Solution

Detailed explanation

We have,

z = 5 x + 4 y

Subject to contraints

y \leq 2 x, x \leq 2 y, x + y \leq 3, x \geq 0, y \geq 0

On taking given constraints as equations, we get the following3 graph.

Intersecting point of line

y = 2 x

and

x + y = 3

A (1, 2)

and intersecting point of line

y = 2 y

and

x + y = 3

B (2, 1)

Here OABO is the required feasible region whose corner points are

O (0, 0), A (1, 2)

and

B (2, 1)

Corner points	$z = 5 x + 4 y$
$O, (0,0)$	$5 \times 0 + 4 \times 0 = 0$
$A (1,2)$	$5 \times 1 + 4 \times 2 = 13$
$B (2,1)$	$5 \times 2 + 4 \times 1 = 14$ (maximum)

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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