If the corner points of bounded feasible region for an LPP are (0,2) (3,0) (6,0) (6,8) and (0, 5) then the minimum value of the objective function f=4x+6y occur at Option 1: (3,0) only Option 2: (0,2) only Option 3: Every point on the line segment joining the points (0,2) and (3,0) Option 4: Only the mid point of the line segment joining the points (0,2) and (3,0)

Answer: Every point on the line segment joining the points (0,2) and (3,0). Solution: The Linear Programming Problem has: Corner points of feasible region: (0,2), (3,0), (6,0), (6,8), (0,5) Objective function: f = 4x + 6y In Linear Programming, the minimum or maximum value of the objective function always occurs at one or more corner points of the feasible region. So instead of checking infinite points, we just plug in the corner points. Evaluating f = 4x + 6y at each corner point:, Corner Point, Calculation, Value of f, (0, 2), 4(0) + 6(2) = 0 + 12, 12, (3, 0), 4(3) + 6(0) = 12 + 0, 12, (6, 0), 4(6) + 6(0) = 24 + 0, 24, (6, 8), 4(6) + 6(8) = 24 + 48, 72, (0, 5), 4(0) + 6(5) = 0 + 30, 30, Comparing all values: 12, 12, 24, 72, 30 The minimum value is 12, and it occurs at two corner points: (0, 2) and (3, 0). When two corner points give the same optimal value, it means the objective function line is parallel to the edge connecting those two points. So every point on the line segment joining (0,2) and (3,0) is also optimal. Therefore, the minimum value of the objective function f = 4x + 6y is 12, occurring at (0, 2) and (3, 0).

Algebra

Linear Programming

Medium

applied

If the corner points of bounded feasible region for an LPP are (0,2) (3,0) (6,0) (6,8) and (0, 5) then the minimum value of the objective function f=4x+6y occur at

(3,0) only

(0,2) only

Every point on the line segment joining the points (0,2) and (3,0)

Only the mid point of the line segment joining the points (0,2) and (3,0)

✅ Correct Option: 3

The Linear Programming Problem has:

Corner points of feasible region: $(0,2)$ , $(3,0)$ , $(6,0)$ , $(6,8)$ , $(0,5)$

Objective function: $f = 4x + 6y$

In Linear Programming, the minimum or maximum value of the objective function always occurs at one or more corner points of the feasible region. So instead of checking infinite points, we just plug in the corner points.

Evaluating $f = 4x + 6y$ at each corner point:

Corner Point	Calculation	Value of $f$
$(0, 2)$	$4(0) + 6(2) = 0 + 12$	$12$
$(3, 0)$	$4(3) + 6(0) = 12 + 0$	$12$
$(6, 0)$	$4(6) + 6(0) = 24 + 0$	$24$
$(6, 8)$	$4(6) + 6(8) = 24 + 48$	$72$
$(0, 5)$	$4(0) + 6(5) = 0 + 30$	$30$

Comparing all values: $12$ , $12$ , $24$ , $72$ , $30$

The minimum value is $12$ , and it occurs at two corner points: $(0, 2)$ and $(3, 0)$ .

When two corner points give the same optimal value, it means the objective function line is parallel to the edge connecting those two points. So every point on the line segment joining $(0,2)$ and $(3,0)$ is also optimal.

Therefore, the minimum value of the objective function $f = 4x + 6y$ is $\boxed{12}$ , occurring at $(0, 2)$ and $(3, 0)$ .

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If the corner points of bounded feasible region for an LPP are (0,2) (3,0) (6,0) (6,8) and (0, 5) then the minimum value of the objective function f=4x+6y occur at

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