What increasing and decreasing mean on an interval, before any calculus

What to assume they know

• Function notation, domain, and the graph of a function, from Class XI
• Interval notation, open and closed, including intervals unbounded on one side
• The graph of a quadratic, and of the sine and cosine
• Comparing two real numbers, and reading an inequality aloud
• The difference between a strict and a non-strict inequality
• What it means to prove a statement for all pairs from a set, rather than for some

What they should be able to do

• Read rising and falling off a graph and off a table of values, and say why neither is a definition
• State all five parts of Definition 1 and identify which inequality each one uses
• Explain what the non-strict inequality in the first two parts permits, and give the function that exercises the permission
• Show that under Definition 1 a constant function satisfies the first, second and third parts at once
• Distinguish the two names for rising and the two names for falling, and use the right one for a stated claim
• Prove a linear function increasing directly from the definition, without any derivative
• Explain why increasing at a single point is defined through an interval around it rather than at the point itself
• Say what "neither increasing nor decreasing" means for a stated function on a stated interval, and reconcile it with the parts of Definition 1
• Identify where the chapter's own Summary states the definition differently from Definition 1, and say which to quote

Where it usually goes wrong

• "Increasing and strictly increasing are two words for one thing." Not in this chapter. Definition 1 gives them different inequalities, and Exercise 6.2 Q11 asks about the strict ones by name. A student who treats them as interchangeable will one day be asked to distinguish them and will have nothing to say.
• "A constant function is obviously neither." Under Definition 1 as printed it is both, and it is also constant. Fig 6.2's third panel says otherwise, and the two cannot both be right. Show the collision rather than hiding it.
• "If the graph goes up somewhere in the interval, the function is increasing on it." The definition quantifies over every pair of inputs from the interval. One pair going the wrong way destroys the claim, which is exactly what happens to the cosine on a full turn.
• "I can settle this by looking at the graph." A graph settles nothing for a function you have not drawn, and the chapter switches to inequalities on Part I p. 152 for that reason. The picture motivates; the inequality proves.
• "Increasing at a point means the graph rises through that point." It means the function is increasing on some open interval containing the point. Definition 2 is a statement about a neighbourhood, and the chapter never once applies it.
• "Checking a few pairs of inputs is enough." It refutes and never establishes. Example 7 works with two unnamed inputs precisely so that the argument covers all pairs at once; a student who tries three numbers has proved nothing.
• "The Summary is a safe place to revise from." For this definition it is not: it prints the strict versions under the non-strict names and adds a derivative condition that contradicts them. Revise from Definition 1 and use the Summary for the tests only.
• "Neither means the function does nothing." It means both behaviours occur inside the interval, so no single verdict holds across it. That is the opposite of doing nothing.

Questions to check understanding

• State all five parts of Definition 1 and name the inequality each uses
• Decide, for a constant function on an interval, which parts of Definition 1 hold
• Prove a linear function increasing directly from the definition, with two unnamed inputs
• Show a stated function neither strictly increasing nor strictly decreasing on a given interval — the form of Exercise 6.2 Q11
• Give the three verdicts for a trigonometric function on three stated intervals — the form of Exercise 6.2 Q3
• Explain why rising at a point is defined through an interval around it
• Given a claim written with a strict or a non-strict sign, say which part of Definition 1 it is

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 own exercises.

• Fig 6.1 and its two tables (§6.3, Part I p. 152). A parabola through the origin opening upward, drawn with a table on each side. Read off the printed page, the left table runs minus two, minus three halves, minus one, minus a half, zero against four, nine quarters, one, one quarter, zero; the right table runs zero, a half, one, three halves, two against zero, one quarter, one, nine quarters, four. Beneath each table sits a sentence about the height of the graph, one saying it falls and one saying it rises. The figure carries a marked point on the horizontal axis with the corresponding height drawn as a dashed segment and labelled. Section 1 is this figure and nothing else.
• The two paragraphs after the figure (Part I p. 152). They read the picture in words, once on each side of the origin, and reach the verdicts for positive and for negative inputs. Then a single sentence announces that a definition in inequalities is coming. That sentence is section 2 of the explanation: the chapter is telling the reader why a picture is not enough, and it is worth stopping on, because the rest of the chapter never argues from a picture again.
• Definition 1, parts (i) and (ii) (Part I p. 152). Read closely, because the extracted text is wrong here. Both parts take two inputs with the first below the second, and both conclude with a non-strict comparison of the two outputs — the first allowing the second output to be at least the first, the second allowing it to be at most. The extraction drops the bar under each sign and prints them as strict; the printed page does not. Every statement of these two parts.
• Definition 1, part (iii) (Part I p. 152). The constant case, stated separately and in the ordinary way — one value for every input.
• Definition 1, parts (iv) and (v) (Part I p. 153). The same two hypotheses as (i) and (ii), with strict comparisons of the outputs, and the word for strictness added to each name. Verified in the same pair of the printed page. Section 5 should set (i) against (iv) with only the sign changing, so the difference is the only thing moving.
• The overlap, worked out (not in the book). Verified: take any function that never moves. For any two inputs with the first below the second, the two outputs are equal, so the second is at least the first and also at most the first. Part (i) is satisfied, part (ii) is satisfied, and part (iii) is satisfied. Under Definition 1 as printed, a constant function is increasing, decreasing and constant at once, and it is neither strictly increasing nor strictly decreasing. This is sections 4 and 6, and it is the pivot of the whole topic.
• Fig 6.2, panels (i), (ii) and (iii) (Part I p. 153). Three small axis pairs. Panel (i), captioned as a strictly increasing function, carries a curve rising from lower left to upper right. Panel (ii), captioned as strictly decreasing, carries a curve falling steeply from the upper left and flattening to the right. Panel (iii), captioned as neither increasing nor decreasing, carries a horizontal segment with a solid dot at each end, drawn above the axis in the first quadrant — a constant function on a closed interval. Read closely precisely to settle what is drawn there. Section 7 is the collision: by the definition on the page above, that segment is increasing and decreasing; by the caption, it is neither. Both readings are on the page.
• Definition 2 (Part I p. 153). Rising or falling at a point, defined by requiring the property on some open interval containing the point. Two things to say: the definition is about a neighbourhood, not about the point, and it is the only definition in §6.3 the chapter never uses again — no example and no exercise item in the chapter asks about a single point. Checked item by item on the printed pages.
• Example 7 (Part I p. 153). The function three less than seven times the input, on the whole line. The question asks the reader to show it increasing; the printed working takes two inputs with the first below the second, multiplies by seven, subtracts three, and reaches a strict comparison of the outputs; the closing line then declares the function strictly increasing. Verified: the working establishes the strict version, which implies the version asked for, so nothing is wrong — but the question and the conclusion use different words, and section 9 should say so plainly rather than let a student think the two are interchangeable.
• Example 9 (Part I pp. 154–155). The cosine, on three intervals. Parts (a) and (b) are derivative arguments and belong to the next topic; part (c) is the one this topic needs. Verified: on the interval from zero to two pi the cosine falls on the first half and rises on the second, so there are two inputs with the first below the second and the outputs rising, and another such pair with the outputs falling; neither part (i) nor part (ii) of Definition 1 can hold, and neither can (iv) or (v). The verdict is the one the chapter reaches.
• Exercise 6.2 Q1 and Q3 (Part I p. 158). Q1 is a linear function to be shown increasing on the whole line — Example 7 with two numbers changed. Verified: two inputs with the first below the second give three times each in the same order, and adding seventeen preserves the order, so the outputs compare strictly. Q3 is the sine on three intervals, matching Example 9's three parts. Verified: on the first quarter turn the sine rises, on the second it falls, and on the half turn it does both, so the third verdict is neither.
• Exercise 6.2 Q11 (Part I p. 159). A quadratic that must be shown to fail both strict claims on a symmetric interval about zero. Verified: the vertex sits at one half, inside the interval, so on the interval the function falls and then rises, and both strict claims fail. Note the item asks about the strict properties by name, which is the only exercise item in §6.3 that does — worth a beat, because it shows the distinction is examinable.
• The Summary's function bullet (Part I p. 185). Read closely. It gives increasing and decreasing on an open interval with strict comparisons of the outputs — Definition 1's parts (iv) and (v) under the names of parts (i) and (ii) — and then adds, under the increasing case only, an alternative in terms of the derivative being at least zero. Section 11 is this bullet against Definition 1. The two disagree, and the bullet also disagrees with itself: a derivative that is allowed to vanish permits a flat stretch, which its own strict comparison on the line above forbids.

Figures to have open

• A redraw of Fig 6.1 (Part I p. 152): an upward parabola through the origin with the two value tables flanking it and the dashed height at a marked point. The chapter's own figure; the explanation needs both tables because section 1 reads values, not shape.
• A redraw of Fig 6.2 panels (i), (ii) and (iii) (Part I p. 153) with the captions exactly as printed. Panel (iii) must be the horizontal segment with a solid dot at each end — it is the whole of section 7 and a redrawn "wiggly" curve would destroy the point.
• A five-row comparison of Definition 1's parts for section 6, name against hypothesis against conclusion. The content is the chapter's own; the layout is added here. Build it with the repo's DataTable component.
• A point with an open interval drawn around it for section 8. Not in the book; Definition 2 is printed with no figure.
• A cosine over one full turn for section 10, with two pairs of inputs marked, one pair rising and one falling. Not in the book; the chapter prints no graph of the cosine.

Where this sits in the book

• NCERT Class 12 Mathematics, Part I, Chapter 6 "Application of Derivatives", §6.3, the opening paragraphs and Fig 6.1, Part I p. 152
• Definition 1, parts (i) to (iii), Part I p. 152; parts (iv) and (v), Part I p. 153
• Fig 6.2, panels (i) to (iii), and Definition 2, Part I p. 153
• Example 7, Part I p. 153; Example 9, part (c), Part I p. 155
• Exercise 6.2, questions 1, 3 and 11, Part I pp. 158–159
• Summary, the increasing-and-decreasing bullet, Part I p. 185