Using the sign of the derivative to split the line into rising and falling intervals

What to assume they know

• Definition 1 and its five parts, from the previous topic
• Differentiating polynomials, quotients, logarithms, exponentials and inverse trigonometric functions, from Chapter 5
• Factorising a quadratic and a cubic, and reading the sign of a product from the signs of its factors
• Solving a trigonometric equation on a stated interval
• Interval notation, and splitting a set at named points
• Continuity at a point, from Chapter 5

What they should be able to do

• State Theorem 1 with the inequalities the page prints, and say why they are the non-strict ones
• Follow the printed proof of the first part and name the result it depends on
• Say where that result now sits in this edition, and what a reader should do about it
• Distinguish Theorem 1 from the Remarks that follow it, which state a related claim with different signs
• Run the standard procedure: differentiate, solve for zero, cut the domain, test each piece
• Build a sign table with one row per piece and read the verdicts off it
• Restrict the procedure to a stated interval and discard cut points that fall outside it
• Handle a derivative that keeps one sign everywhere, and say what that settles
• Handle a derivative that vanishes at a point without changing sign
• Report intervals with the bracket shapes the chapter uses, and say when an endpoint may be included

Where it usually goes wrong

• "Theorem 1 needs the derivative to be positive." As printed it needs the derivative to be at least zero. The strict version is in the Remarks four lines below and in the proof above, and confusing the three is easy because the page does not flag the difference. Read the theorem off the page image, not off any extraction.
• "The cut points belong to one of the pieces." They are the boundaries. The chapter reports open pieces in Examples 10 and 11 and only closes them in Example 12, after a separate continuity argument. A student who silently includes an endpoint has skipped that argument.
• "Every zero of the derivative cuts the domain into a rising piece and a falling piece." Exercise 6.2 Q6(e) has a derivative vanishing at three inputs and changing sign at only one of them. The other two are zeros that change nothing.
• "A sign table is the answer." It is bookkeeping. The exercise's "prove that" items — seven of them — want the argument written out. A table handed in where a proof was asked for scores nothing.
• "If the derivative is a mess I have to expand it." Almost every item here factors, and the factored form is what makes the sign readable. Miscellaneous Example 33 factors a quartic's derivative into three linear pieces and the whole problem collapses.
• "Testing one number inside a piece is not rigorous." It is exactly what the chapter does, in a parenthesis, twice, on Part I p. 180 — and it is valid because the derivative is a continuous function with no zero inside the piece. Say why it works rather than banning it.
• "The theorem is about a closed interval, so I cannot use it on an open one." The Remarks extend it, and Example 9 relies on that extension without saying so. Point at the Remarks when the interval is open.
• "I can look up the Mean Value Theorem in Chapter 5." In this edition you cannot. See Notes, and tell the student where the statement they need actually is.

Questions to check understanding

• State Theorem 1 with its hypotheses and the inequality in each conclusion
• Find the intervals on which a stated polynomial rises and falls, and present them in a sign table
• Restrict the same procedure to a stated interval and justify discarding a solution outside it
• Prove a stated function rising on its whole domain by showing its derivative keeps one sign — the form of Exercise 6.2 Q7 to Q10
• Find the values of a parameter that make a stated function rise on a stated interval — the form of Exercise 6.2 Q14
• Choose the interval on which a stated function rises, from four options — the form of Exercise 6.2 Q19
• Reproduce the proof that a positive derivative forces rising — the form of Miscellaneous Exercise Q13

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.

• Theorem 1 (§6.3, Part I p. 153). Read closely, because the extracted text is wrong here. The function is assumed unbroken on a closed interval and differentiable inside it. The three conclusions are then: the function rises when the derivative is at least zero throughout the inside; it falls when the derivative is at most zero throughout; and it is constant when the derivative is exactly zero throughout. The extraction drops the bar under the first two signs. The non-strict form is the one that matches Definition 1's non-strict parts, which is why it is worth reading twice.
• The printed proof of part (a) (Part I p. 154). Four lines: take two inputs in order; apply the named result from Chapter 5 to get the difference of outputs as the derivative at some in-between point times the difference of inputs; conclude the difference of outputs is positive, with a parenthesis citing that the derivative there is positive; conclude the second output exceeds the first. Note that the parenthesis says the derivative is positive, where the theorem above assumed only that it is at least zero. Both lines were read on the two facing pages. The proof establishes the strict conclusion from a strict hypothesis and is then used to close the non-strict claim. Section 3 should present the proof and section 4 the gap.
• The result the proof depends on (Part I pp. 153 and 154). The chapter names it twice, once in the sentence introducing Theorem 1 and once inside the proof, where it is given a theorem number and attributed to Chapter 5. See Notes: that number is not reached by Chapter 5 in this edition.
• The Remarks (Part I p. 154). A more general statement: a positive derivative on the inside of an interval, together with continuity on the interval, gives a rising function, and likewise for negative and falling. This is a third set of signs in two pages — non-strict in Theorem 1, strict in its proof, strict again here. Section 5 should lay all three side by side rather than pick one.
• Example 8 (Part I p. 154). A cubic whose derivative is a quadratic. The chapter completes the square and shows the derivative is one more than three times a square, so it is positive everywhere. Verified: three times the square of one less than the input, plus one, which is at least one for every real input. This is the model for section 11's first half: no cut points at all, because the derivative never reaches zero.
• Example 9 (Part I pp. 154–155). The cosine on three intervals, with the derivative equal to the negated sine. Verified: on the first half turn the sine is positive so the derivative is negative and the cosine falls; on the second half turn the sine is negative so the derivative is positive and the cosine rises; over the full turn both happen, so neither verdict holds. Note that this example applies the theorem on open intervals, where the theorem as printed is about a closed one — the Remarks are what license it.
• Example 10 and Fig 6.3 (Part I p. 155). A parabola; its derivative vanishes at one input, cutting the line into two pieces, and the sign is read on each. Verified: the derivative is two less than twice the input, negative below two and positive above, so the function falls on the lower piece and rises on the upper. Fig 6.3 is a horizontal line with arrowheads at both ends, the two ends labelled with the signed infinities, and a single tick at the cut point. Section 7 is this example and this figure.
• Example 11, Fig 6.4 and the first table (Part I pp. 155–156). A cubic whose derivative factors into twelve times two linear factors, vanishing at minus two and at three. Verified: on the lowest piece both factors are negative so the product is positive; on the middle piece the signs differ so the product is negative; on the top piece both are positive. The function rises, falls, rises. The chapter then prints a three-row table with columns for the piece, the sign, and the verdict, and closes by saying the function is neither rising nor falling on the whole line. Fig 6.4 is the same number line as Fig 6.3 with two ticks. Section 8 is this example, and the table is the section's whole point.
• Example 12, Fig 6.5 and the continuity step (Part I pp. 156–157). A sine of three times the input, restricted to a quarter turn. The derivative vanishes where the tripled input is a right angle, which inside the stated interval happens once. Verified: the tripled input runs from zero to three right angles as the input crosses the quarter turn, so the derivative is positive on the first sixth of a half turn and negative after it. The chapter first states the verdicts on half-open pieces and then, in a separate step at the top of the next page, invokes continuity at the two endpoints to upgrade them to closed ones. That upgrade step is the only place in the chapter where the continuity hypothesis does visible work, and section 10 exists for it.
• Example 13, Fig 6.6 and the second table (Part I pp. 157–158). The sum of sine and cosine over a full turn; the derivative vanishes where the two are equal. Verified: that happens at an eighth of a turn and at five eighths, cutting the full turn into three pieces, on which the function rises, falls and rises. The second printed table has the same three columns as the first but records only the sign, not the factorisation. Fig 6.6 is a number line with the two interior cut points and both endpoints marked.
• Exercise 6.2 Q4, Q5 and Q6 (Part I p. 158). The core drill: a quadratic, a cubic, and five more functions. Verified: Q4's derivative vanishes at three quarters, so the function falls below and rises above. Q5's derivative factors with zeros at minus two and three, so the function rises, falls, rises. Q6(a) turns at minus one, rising above; (b) turns at minus three halves, falling above; (c) has a derivative that is negative six times a product of two linear factors, so it rises only between minus two and minus one; (d) turns at minus nine halves, falling above; (e) has a derivative equal to six times the input less one, times two squared factors, so it falls below one and rises above, with the derivative also vanishing at minus one and at three without changing sign there.
• Exercise 6.2 Q7, Q8, Q9, Q10 and Q16 to Q18 (Part I pp. 158–159). Seven "prove that" items. Verified for Q8: the derivative is four times the input, times one less than the input, times two less than the input, so the function rises between zero and one and again above two. Verified for Q10: the derivative of the logarithm is the reciprocal of the input, positive throughout the stated domain. The others follow the same pattern.
• Exercise 6.2 Q12, Q13 and Q19 (Part I p. 159). The three multiple-choice items. Verified for Q13: the derivative is a hundred times the ninety-ninth power plus the cosine; on each of the three offered intervals the power term dominates and the derivative stays positive, so the function decreases on none of them and the fourth option is right. Verified for Q19: the derivative is the input times the exponential of its negative, times two less the input, so it is positive exactly between zero and two, which is the fourth option. Q12 needs a warning and gets one in Notes.
• Exercise 6.2 Q14 and Q15 (Part I p. 159). Two items where the answer is a condition rather than an interval. Verified for Q14: the derivative is twice the input plus the parameter, and it must be non-negative across the whole stated interval, which forces the parameter to be at least minus two. Verified for Q15: the derivative is one less the reciprocal of the square of the input, which is non-negative exactly when the square is at least one, so the function rises on any interval that avoids the closed interval between minus one and one.
• Miscellaneous Example 33 and Fig 6.22 (Part I pp. 179–180). A quartic whose derivative simplifies to six fifths times a product of three linear factors, vanishing at minus two, one and three. Verified: the four pieces give falling, rising, falling, rising in that order. The chapter works each piece by naming the sign of each factor and then, in a parenthesis, checks the verdict at one convenient input. That parenthesis is the practical technique the chapter never states as a rule, and section 6 should promote it: pick a test input inside the piece.
• Miscellaneous Example 34 (Part I pp. 180–181). An inverse tangent of a sum of sine and cosine, to be shown rising on an eighth of a turn. Verified: the derivative simplifies to the difference of cosine and sine over two plus the sine of the doubled input, whose denominator is positive on the interval, so the sign is decided by the numerator, which is positive below an eighth of a turn.
• Miscellaneous Exercise Q3 and Q4 (Part I p. 183). Verified for Q3: the derivative simplifies to the cosine times four less the cosine, over the square of two plus the cosine; the second factor and the denominator are positive always, so the sign follows the cosine alone — rising where the cosine is positive and falling where it is negative. The item states no interval; see Notes. Verified for Q4: the derivative is three times the input squared less three over the input to the fourth, which shares the sign of one less than the sixth power, so the function rises when the input is outside the closed interval between minus one and one, and falls inside it on either side of zero.
• Miscellaneous Exercise Q13 (Part I p. 184). The item asks the student to prove that a positive derivative throughout an open interval makes the function increasing on it. That is Theorem 1's first part with a strict hypothesis — the proof on Part I p. 154, reproduced. Section 12 is this item, and it is where the missing dependency bites hardest.

Figures to have open

• Redraws of Fig 6.3 and Fig 6.4 (Part I pp. 155–156): a horizontal line with arrowheads at both ends, labelled with the signed infinities, carrying one tick and two ticks respectively. The chapter's own figures, and the explanation needs both because the second is the first with one more cut.
• Redraws of Fig 6.5 and Fig 6.6 (Part I pp. 156–157): the same device on a bounded interval, with the endpoints marked as well as the interior cut points. The chapter's own; they matter because they show the endpoints labelled, which Fig 6.3 and Fig 6.4 do not.
• A redraw of Fig 6.22 (Part I p. 180): a number line with three interior cut points. The chapter's own.
• A three-column sign table for section 8, with a row per piece. The content and the column headings are the chapter's own from Part I p. 156. Build it with the repo's DataTable component.
• A three-column comparison for section 5 of the three sign conventions in Theorem 1, its proof and the Remarks. Not in the book.
• The graph of a cubic with a derivative that never vanishes, for section 11. Not in the book; the chapter prints no graph in §6.3 other than the parabola of Fig 6.1.

Where this sits in the book

• NCERT Class 12 Mathematics, Part I, Chapter 6 "Application of Derivatives", §6.3, Theorem 1, Part I p. 153
• The proof of Theorem 1 part (a), and the Remarks, Part I p. 154
• Examples 8 to 13, with Fig 6.3 to Fig 6.6 and the two tables, Part I pp. 154–158
• Exercise 6.2, questions 2 and 4 to 19, Part I pp. 158–159
• Miscellaneous Examples 33 and 34, with Fig 6.22, Part I pp. 179–181
• Miscellaneous Exercise on Chapter 6, questions 3, 4 and 13, Part I pp. 183–184