Board 2025 Class 12 Maths Q14 Solution: Maximum and minimum of Z = 18x + 9y at given corner points

The question

The corner points of the feasible region in graphical representation of a L.P.P. are $(2, 72)$ ⁠, $(15, 20)$ and $(40, 15)$ ⁠. If $Z = 18x + 9y$ be the objective function, then

(A) Z is maximum at $(2, 72)$ ⁠, minimum at $(15, 20)$

(B) Z is maximum at $(15, 20)$ minimum at $(40, 15)$

(C) Z is maximum at $(40, 15)$ ⁠, minimum at $(15, 20)$

(D) Z is maximum at $(40, 15)$ ⁠, minimum at $(2, 72)$

Exam answer

Corner point method: find Z at each corner point.

Why?

By the corner point theorem, when $Z = ax + by$ has a maximum or a minimum over the feasible region, it occurs at a corner point. So we only need Z at the three corners.

\begin{aligned} Z(2, 72) &= 36 + 648 = 684 \\ Z(15, 20) &= 270 + 180 = 450 \\ Z(40, 15) &= 720 + 135 = 855 \end{aligned}

Why?

Put x and y of each point into $Z = 18x + 9y$ ⁠:

\begin{aligned} 18 \times 2 = 36,\ &9 \times 72 = 648 \\ 18 \times 15 = 270,\ &9 \times 20 = 180 \\ 18 \times 40 = 720,\ &9 \times 15 = 135 \end{aligned}

Comparing, $450 < 684 < 855$ ⁠.

Maximum $Z = 855$ at $(40, 15)$ ⁠; minimum $Z = 450$ at $(15, 20)$ ⁠.

Why?

The largest corner value is the maximum and the smallest is the minimum.

Watch out: the point with the biggest coordinate is not automatically the maximum. $(2, 72)$ has the largest y, but Z there is only 684.

Answer: (C) Z is maximum at $(40, 15)$ ⁠, minimum at $(15, 20)$

Where this comes from

Chapter 12, Linear Programming: Corner point method.

Question type: Both maximum and minimum, and a tie along an edge

How to spot it: click to see

Type 3. Both maximum and minimum, and a tie along an edge

Spot it: "Minimise and maximise $Z$ ". Often one rule, like $x - 2y \ge 0$ ⁠, is a line through the origin.

Here: The three corners are given, so go straight to one table: $Z = 684, 450, 855$ ⁠. The largest, 855 at $(40, 15)$ ⁠, is the maximum and the smallest, 450 at $(15, 20)$ ⁠, the minimum.

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Q14: Maximum and minimum of Z=18x+9yZ = 18x + 9yZ=18x+9y at given corner points

Q14: Maximum and minimum of $Z = 18x + 9y$ at given corner points