Board 2023 Class 12 Maths Q17 Solution: Maximum and minimum of z = 18x + 9y at three corner points

The question

The corner points of the feasible region in the graphical representation of a linear programming problem 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

Value of $z = 18x + 9y$ at each corner point:

Why?

This is the corner point method. If z has a maximum or a minimum value on 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?

Substitute each point into $18x + 9y$ ⁠. For example, at $(2, 72)$ ⁠:

\begin{aligned} &18(2) + 9(72) \\ &= 36 + 648 \\ &= 684 \end{aligned}

Similarly $18(15) = 270$ ⁠, $9(20) = 180$ ⁠, $18(40) = 720$ and $9(15) = 135$ ⁠.

\begin{aligned} &\text{At } (15, 20)\text{: } 18(15) + 9(20) \\ &= 270 + 180 = 450 \\[4pt] &\text{At } (40, 15)\text{: } 18(40) + 9(15) \\ &= 720 + 135 = 855 \end{aligned}

So z is maximum (855) at (40, 15) and minimum (450) at (15, 20).

Why?

The largest of the three values is the maximum and the smallest is the minimum.

Watch out: $(2, 72)$ has the biggest coordinate, but $y$ has the smaller coefficient 9, so it does not give the maximum. Options (a) and (d) fall into this.

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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Q17: Maximum and minimum of z=18x+9yz = 18x + 9yz=18x+9y at three corner points

Q17: Maximum and minimum of $z = 18x + 9y$ at three corner points