The objective function of an LPP is $z = αx + βy, (α,β > 0)$ that has to be maximized/minimized subject to constraints $x + y ≤2, x≥0, y ≥0$, Then $\text{max (z)- min (z)}$ is equal to

$2\, max\{α,β\}$

The correct answer is Option (1) → $2\, max\{α,β\}$

Given objective function:

$z = \alpha x + \beta y, \; (\alpha, \beta > 0)$

Subject to constraints:

$x + y \le 2, \; x \ge 0, \; y \ge 0$

The feasible region is bounded by the triangle with vertices:

$(0,0), (2,0), (0,2)$

Compute $z$ at each vertex:

At $(0,0)$: $z_1 = 0$

At $(2,0)$: $z_2 = 2\alpha$

At $(0,2)$: $z_3 = 2\beta$

Maximum value: $\max(2\alpha, 2\beta)$

Minimum value: $0$

Hence,

$\max(z) - \min(z) = 2 \max(\alpha, \beta)$

Therefore, $\max(z) - \min(z) = 2\max(\alpha, \beta)$