Why an optimum, if there is one, has to sit at a corner

What to assume they know

• The feasible region as the overlap of the half planes the constraints allow, from the previous topic
• Finding where two lines cross by solving their equations together
• Substituting a pair of numbers into a linear expression
• What it means for a set of points to be bounded, and the chapter's circle test
• The idea of a straight line of constant value, at the level of recognising that one equation can be satisfied by infinitely many pairs

What they should be able to do

• State the difficulty the chapter names before it introduces its two theorems, and say why an exhaustive search cannot settle it
• Define a corner point in the chapter's own terms, and identify the corners of a drawn region, remembering that the axes are boundary lines
• Explain the sliding-line picture, and use it to say why the last point of contact with a region is a corner or an edge
• State Theorem 1 and say precisely what it claims and what it does not
• State Theorem 2 and say which extra thing it adds, and under what hypothesis
• State the Remark for the unbounded case and say which of the two theorems survives it
• Say that both results are given without proof and that the chapter says so
• Find the corners of a bounded region, evaluate the objective at each, and read the answer off the list
• Recognise a tie between two corners, and explain why every point of the segment joining them is then also optimal
• Identify, from the algebra alone, when a tie is going to happen

Where it usually goes wrong

• "Theorem 1 guarantees there is an answer." It does not. It says where an optimal value would have to be, if there is one. The existence claim is Theorem 2's and it needs the region to be bounded. Exercise 12.1 item nine has a region with corners and no maximum at all.
• "The corners are only where two constraint lines cross." The two axes are boundary lines too, because non-negativity is a pair of constraints. The origin is a corner of the running problem's region, and dropping it is the commonest way to lose a candidate.
• "An optimum can sit in the middle of the region." It cannot, unless the objective is constant everywhere. Pushing a line of constant value across the region always leaves the interior before it leaves the boundary. That is the content of Theorem 1 and it is why the search is finite.
• "The optimum sits at exactly one corner." It can sit at two, and then it sits at every point in between. The chapter's own Example 3 does this, and three of the ten exercise items do.
• "If two corners tie, I pick whichever I like and the others are wrong." Both are right, and so is every point of the segment joining them. A question asking for the optimal solution has been badly worded whenever a tie occurs; the honest answer names the segment.
• "Theorem 2 says the maximum and the minimum are at different corners." It says each is at a corner. Nothing forbids the same corner, and nothing forbids either from being shared between two.
• "The chapter proved these results." It states in one sentence that the proofs are outside its scope, and the chapter contains no proof of anything. An explanation that narrates the theorems in a proving voice teaches a student to expect a justification they will never be shown.
• "An unbounded region has no corners, so the theorems do not apply." It has corners — the chapter's own unbounded figure has four — and Theorem 1 applies to it in full. What is lost is only the guarantee that an optimum exists.
• "A tie is a coincidence." In every instance this chapter contains, it is a consequence of the objective being a positive multiple of a constraint expression, and it can be spotted in the algebra before any drawing.

Questions to check understanding

• List the corner points of a drawn region, including any that lie on an axis
• State what Theorem 1 claims and what it does not, in one sentence each
• Say which theorem is doing the work in a given situation, given whether the region is bounded
• Evaluate the objective at every corner of a bounded region and name the optimum
• Explain why an optimum cannot sit strictly inside the region
• Given two corners returning the same value, say what else is optimal and justify it
• Predict from the algebra alone whether a stated problem will have multiple optimal solutions — the form Exercise 12.1 items six, seven and eight support
• Show that the smallest value of a stated problem is attained at more than two separate points — the form of Exercise 12.1 item six

Examples worth working on the board

Values marked verified are worked out here from the chapter's own printed data; neither answers file was opened, and this chapter prints no answers to its exercises.

• The difficulty, stated by the chapter (§12.2.2, Part II p. 398). Before either theorem the chapter observes that every point of the region satisfies every constraint, that there are infinitely many of them, and that it is therefore not obvious how to find the one that makes the objective largest. Open the explanation on this. The two theorems are answers to a question, and a student who never heard the question receives them as decoration.
• The corner-point footnote (Part II p. 398). A corner point is a point of the region where two boundary lines cross. Verified reading: the two axes are boundary lines, because the two non-negative conditions are constraints like any other — which is why the origin is a corner of the running problem's region. The chapter never spells that out and the omission costs students two corners on almost every drawing.
• The sliding-line picture (entirely added here). Fix a value for the objective and the pairs giving that value lie on a straight line; change the value and the line moves parallel to itself. Push it across the region in the improving direction and the last thing it touches before leaving is a corner — or, if it happens to be parallel to a boundary, a whole edge. Verified on the chapter's own running problem: the objective of the furniture problem is constant along lines of one fixed slope, and pushing such a line up and to the right across Fig 12.1's quadrilateral leaves it last at the corner ten and fifty. The chapter offers no intuition of any kind for its two theorems, and the words for this device — level line, objective line — occur nowhere in it. This section is the whole reason the topic exists.
• Theorem 1 (Part II p. 398). Its content, in the wording used here: if the objective has an optimal value on the region, that value is attained at a corner. Verified reading of the logical form: it is conditional. It promises a location and says nothing whatever about existence. Students read it as guaranteeing an answer; it does not.
• Theorem 2 (Part II p. 398). Its content: when the region is bounded, the objective has both a largest and a smallest value on it, and each is attained at a corner. Verified reading: this is the existence half, and it is bought entirely by the word bounded. Set the two theorems side by side — one about where, one about whether — because the chapter prints them as a pair of equals and they do quite different jobs.
• The Remark (Part II p. 398). When the region is unbounded, a largest or a smallest value may fail to exist; but if one exists it is still at a corner, and the chapter attributes that surviving half to Theorem 1 by name. Verified: that attribution is correct, because Theorem 1 was conditional and its hypothesis is exactly an optimal value exists. This is the cleanest piece of logic in the chapter and it is worth showing as such.
• The chapter says the proofs are not here (Part II p. 398). One sentence, immediately before Theorem 1, placing both proofs outside the book's scope. Verified across the extracted text of all twelve folios: the word prove occurs zero times in the chapter and proofs once, in that sentence. Nothing in this chapter is proved. Say so; it is a mark of honesty rather than a gap, and it tells a student where the sliding-line picture stands — as an argument for plausibility, not a proof.
• The furniture problem finished (Part II p. 398). The corners are the origin; twenty and zero; ten and fifty; and zero and sixty. Verified: the objective takes the values zero, five thousand, six thousand two hundred and fifty, and four thousand five hundred at those four points respectively, so the largest is six thousand two hundred and fifty, at ten and fifty. Part II p. 399 reads that back as ten tables and fifty chairs. A layout warning: the sentence naming the corners on Part II p. 398 lists them in one order and the table directly beneath it lists them in another. Both are right; a student checking one against the other will think one is wrong. Put them in one order.
• Example 3 and Fig 12.4 (Part II p. 401). A bounded four-cornered region with corners at zero and ten, five and five, fifteen and fifteen, and zero and twenty; the problem asks for both a smallest and a largest value. Verified: the objective takes ninety, sixty, one hundred and eighty, and one hundred and eighty at those four corners, so the smallest is sixty at five and five, and the largest is one hundred and eighty attained at two corners. The chapter's own table marks the tie with a brace and labels it as multiple optimal solutions. All four boundary lines in the figure are drawn solid; read off a three-hundred-dot-per-inch the printed page.
• Why a tie spreads along the whole segment (Remark, Part II p. 401, argued by the explanation). The chapter asserts that every point between the two tied corners gives the same value and does not say why. Verified: the objective is a linear expression, so along a straight path between two points its value moves steadily from the value at one end to the value at the other; if the two ends agree, it never moves at all. One line of reasoning, and it converts an assertion into something a student can reconstruct.
• Seeing the tie coming (not in the book). A tie happens exactly when the objective is a positive multiple of one of the constraint expressions, because then the objective is constant along that constraint's own boundary line. Verified: in Example 3 the objective is three times the expression in the first constraint, whose limit is sixty — and three sixties is the hundred and eighty the tie returns. In Exercise 12.1 item six the objective is the expression of the second constraint, whose limit is six, and the tied value is six. In item seven the objective is five times the expression of the first constraint, whose limit is a hundred and twenty, and the tied value is six hundred. In item eight the objective is again the expression of the first constraint, whose limit is a hundred, and the tied minimum is a hundred. Four instances, four matches. The chapter never mentions this and it turns a surprise into something predictable from the algebra before anything is drawn.
• Exercise 12.1 items six, seven and eight (Part II p. 404). Verified by working added here: item six asks for a minimum on an unbounded region and its own instruction line asks the student to show that the smallest value is attained at more than two separate points — the corners are zero and three, and six and zero, both returning six, and the whole segment between them returns six. Item seven is bounded with corners at sixty and zero, a hundred and twenty and zero, sixty and thirty, and forty and twenty; the values are three hundred, six hundred, six hundred and four hundred, so the smallest is three hundred at one corner and the largest is six hundred at two. Item eight is bounded with corners at zero and fifty, twenty and forty, fifty and a hundred, and zero and two hundred; the values are a hundred, a hundred, two hundred and fifty, and four hundred, so the smallest is a hundred at two corners and the largest four hundred at one. These three are this topic's practice set.
• The general features listed at the end (Remarks, items (i) and (ii), Part II p. 403). Item (i) asserts convexity. Item (ii) restates the corner result and adds the tie rule. Verified: item (ii) says nothing Theorem 1, Theorem 2 and the Part II p. 401 Remark had not already said — it is a summary, and it is the closest thing the chapter has to a Summary bullet about solving, since the printed Summary on Part II p. 404 contains none.

Figures to have open

• A region with points igniting endlessly for section 1. Entirely added here; the chapter states the difficulty in prose and draws nothing for it.
• A corner-finding pass over Fig 12.1's region (Part II p. 397) for section 2, with the two axes traced in the same weight as the two constraint lines so the origin is visibly a crossing like any other. The region is the chapter's; the treatment is added here.
• A sliding line of constant value for section 3, over the same region, with a running value readout. Entirely added here — the chapter draws no such line anywhere and names no such device.
• A two-column theorem panel for section 5. The content is the chapter's Part II p. 398; the side-by-side framing is added here and is the largest editorial decision in this brief.
• A four-row corner table for section 8, built with the repo's DataTable component. The numbers are the chapter's own Part II p. 398 table; the row order is added here, and it should match the order the corners are named in the sentence above that table rather than the order the table itself prints.
• A redraw of Fig 12.4 (Part II p. 401) for section 9, with the two tied corners and the segment between them. Read the corner coordinates off the printed page listed in Notes; all four boundary lines are solid and the region is bounded.
• A four-row algebra panel for section 10 pairing each tie case with the constraint its objective is a multiple of. Entirely added here.

Where this sits in the book

• NCERT Class 12 Mathematics, Chapter 12 "Linear Programming", §12.2.2, the statement of the difficulty and the sentence placing both proofs outside the book, Part II p. 398
• Theorem 1, Theorem 2 and the Remark on the unbounded case, with the two footnotes defining a corner point and the bounded case, Part II p. 398
• The corner table for the running problem, Part II p. 398, and its reading back into tables and chairs, Part II p. 399
• Example 3 with Fig 12.4 and its corner table, and the Remark on multiple optimal solutions, Part II p. 401
• Remarks, items (i) and (ii), Part II p. 403
• Exercise 12.1, items six, seven and eight, Part II p. 404