Refer to Exercise 32. Maximum of $F$ - Minimum of $F =$ (A) $60$ (B) $48$ (C) $42$ (D) $18$

Refer to Exercise 32. Maximum of F - Minimum of F – Class 12 Mathematics

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Concept understanding — Corner Point Theorem

The Corner Point Theorem: Why the Best Answer Hides at the Edges

Imagine maximising profit for a factory that makes two products, with limited raw materials, machine hours, and labour. Every combination that doesn't break a limit is a feasible solution. Plot them all on a graph and they form a shape — always a polygon if your constraints are straight lines.

Where is the best (maximum profit) point? You might think anywhere inside the shape. But the Corner Point Theorem says something surprising: the best point is always at a corner — a vertex of the polygon. Never floating in the middle of an edge or inside.

Note

This theorem is the backbone of linear programming — the method for solving optimisation problems with straight-line constraints.

The Intuition: Why Corners Win

Think of profit as a line you slide across the polygon; each position represents a profit level. You push the line as far as possible (higher profit) while still touching the polygon. The last point of contact before the line escapes is always a corner.

Why? Because both the profit line and the polygon's edges are straight, and the farthest point in any straight-line direction from a polygon is always a vertex. This holds for any flat-sided shape.

Tip

Solving a linear programming problem by hand, you only need to check the corners — usually just 3–5 points, not the infinite points inside.

The Precise Statement

Corner Point Theorem (Fundamental Theorem of Linear Programming): If a linear programming problem has an optimal solution, then that optimal solution occurs at at least one corner point (vertex) of the feasible region.

Three key parts:

"If it has an optimal solution" — sometimes the problem is unbounded (profit increases forever) or infeasible (no point satisfies all constraints). The theorem applies only when a best answer exists.
"At least one corner point" — several corners can give the same optimal value. If the profit line is parallel to an edge, every point on that edge is optimal, including both endpoints (corners).
"Of the feasible region" — the polygon formed by all constraints. Corners are where two constraint lines intersect.

Why This Matters for Exams

To solve a linear programming problem:

Find all corner points (solve pairs of constraint equations).
Plug each corner into the objective function.
Pick the best value.

The theorem guarantees you haven't missed a better answer hiding in the middle. …

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Evaluating $F=4x+6y$ at the corner points gives a maximum of $72$ and a minimum of $12$ , so Max $-$ Min $=60$ — option (A).

What the question refers to

This item follows on from the earlier exercise whose feasible region has the corner points

(0,2),\quad (3,0),\quad (6,0),\quad (6,8),\quad (0,5),

with objective function $F = 4x + 6y$ . By the Corner-Point Theorem, the largest and smallest values of a linear objective over a feasible region are always found at these vertices, so we simply test each one.

Evaluate F at every corner

Corner $(x,y)$	$F = 4x + 6y$
$(0,2)$	$0+12 = 12$
$(3,0)$	$12+0 = 12$
$(6,0)$	$24+0 = 24$

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Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.

Showing the 12 most recent of 14 on this concept.

GUJCET 2026Set x1 markMCQ
$(0,10), (5,5), (15,15), (0,20)$
›Reveal solutionSolution
[!TLDR] The minimum of $z=3x+9y$ over the corner points is $60$ at $(5,5)$ .

Concept

In linear programming, a linear objective attains its optimum over a bounded feasible region at a corner (vertex). So evaluate $z$ at every corner point and choose the smallest for the minimum.

Solution …
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GSEB Higher Secondary Certificate (HSC) Examination 2026Set ANNUAL1 markMCQ
$Z=510x+675y$
(a) 1,72,800
(b) 1,62,000
(c) 1,53,000
(d) 1,70,000
›Reveal solutionSolution

Evaluate $Z$ at each corner point and pick the largest.

$Z=510x+675y$ : $(0,0)\to0$ ; $(300,0)\to153000$ ; $(180,120)\to91800+81000=172800$ ; $(0,240)\to162000$ .

…

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GUJCET 2025Set 031 markMCQ
$(0, 0)$
›Reveal solutionSolution
For a bounded LP region the optimum sits at a corner; test all corners.

With $z=40x+30y$ :

$(0,0)\to0$

$(0,40)\to1200$

$(20,40)\to800+1200=2000$ …
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GSEB Higher Secondary Certificate (HSC) Examination 2025Set ANNUAL1 markMCQ
$Z = px + qy$
(a) $p = 3q$
(b) $q = 2p$
(c) $q = 3p$
(d) $p = 2q$
›Reveal solutionSolution

Plug both corner points into $Z=px+qy$ to get two equations in $p,q$ , then solve.

At $(0,10)$ : $10q=90 \Rightarrow q=9$ .

…

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GSEB Higher Secondary Certificate (HSC) Examination 2025Set ANNUAL1 markMCQ
$Z = -50x + 20y$
(a) $-100$
(b) $-300$
(c) $-200$
(d) $-500$
›Reveal solutionSolution

Evaluate $Z$ at every corner point of the bounded feasible region; the minimum is the smallest value.

$Z(0,5)=0+100=100$

$Z(0,3)=0+60=60$

$Z(1,0)=-50+0=-50$

$Z(6,0)=-300+0=-300$

…

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GUJCET 2024Set 131 markMCQ
$(0, 6), (3, 3), (9, 9), (0, 12)$
›Reveal solutionSolution
Evaluate $z=6x+12y$ at every corner point; the maximum is the answer.

Steps.

$(0,6):\ 0+72 = 72$

$(3,3):\ 18+36 = 54$

$(9,9):\ 54+108 = 162$ …
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GSEB Higher Secondary Certificate (HSC) Examination 2024Set ANNUAL1 markMCQ
$Z = 3x + 2y$
(a) $36$
(b) $46$
(c) $13$
(d) $56$
›Reveal solutionSolution

In an LP problem the optimum of a linear objective function always occurs at a corner point of the feasible region.

Evaluate $Z=3x+2y$ at each corner point:

$(12,0): Z=36$

$(4,2): Z=12+4=16$

$(1,5): Z=3+10=13$ …

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GUJCET 2023Set 091 markMCQ
$(0, 10)$
›Reveal solutionSolution
[!TLDR]

$[L]=M^1L^2T^{-2}A^{-2}$ .

Concept

Self inductance relates to stored magnetic energy $U=\tfrac12 L I^2$ (NCERT/GSEB electromagnetic induction), so $L=\dfrac{2U}{I^2}$ .

Solution

Energy has dimensions $[ML^2T^{-2}]$ and current $[A]$ : …
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GUJCET 2022Set 081 markMCQ
$Z=3x+9y$
›Reveal solutionSolution
In LP the optimum of a linear objective occurs at a corner point — just test them all.

Concept. Evaluate $Z=3x+9y$ at every corner and pick the smallest.

Solution.

$(0,10):\;0+90=90$

$(5,5):\;15+45=60$ …
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GSEB Higher Secondary Certificate (HSC) Examination 2022Set ANNUAL1 markMCQ
$2x + y \leq 10$
(a) $q = p$
(b) $q = 2p$
(c) $q = 3p$
(d) $p = 3q$
›Reveal solutionSolution

If the maximum of $Z$ occurs at two adjacent corner points, $Z$ is equal there.

$Z = qx+py$ . Set $Z(3,4) = Z(0,5)$ :

…

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GSEB Higher Secondary Certificate (HSC) Examination 2022Set ANNUAL1 markMCQ
$(2, 72)$
(a) $(2, 72)$
(b) $(15, 20)$
(c) $(40, 15)$
(d) $(0, 0)$
›Reveal solutionSolution

Evaluate the objective at each corner point and pick the smallest.

$Z(2,72) = 12+216 = 228.$

$Z(15,20) = 90+60 = 150.$

$Z(40,15) = 240+45 = 285.$

…

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GSEB Higher Secondary Certificate (HSC) Examination 2020Set ANNUAL1 markMCQ
$O(0,0), A(10,0), B(0,20), C(15,15)$
(a) 30
(b) 130
(c) -120
(d) -370
›Reveal solutionSolution

Evaluate the objective function at all given vertices of the feasible region and pick the smallest.

$Z=10x-20y+30$ at each vertex:

$O(0,0)$ : $Z=30$ .

$A(10,0)$ : $Z=100+30=130$ .

$B(0,20)$ : $Z=-400+30=-370$ .

…

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Q.Refer to Exercise 32. Maximum of $F$ - Minimum of $F =$
(A) $60$
(B) $48$
(C) $42$
(D) $18$

Concept understanding — Corner Point Theorem

The Corner Point Theorem: Why the Best Answer Hides at the Edges

The Intuition: Why Corners Win

The Precise Statement

Why This Matters for Exams

What the question refers to

Evaluate F at every corner

Method: Extracting (max $-$ min) or (max $+$ min) from a feasible region

Steps

Common Mistakes

Concept

Solution …

Concept

Solution

Q.Refer to Exercise 32. Maximum of FFF - Minimum of F=F =F= (A) 606060 (B) 484848 (C) 424242 (D) 181818

What the question refers to

Evaluate F at every corner

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Q.Refer to Exercise 32. Maximum of $F$ - Minimum of $F =$
(A) $60$
(B) $48$
(C) $42$
(D) $18$

Unlock everything free for 14 days