A sign change either side of a critical point decides which kind it is

What to assume they know

• Critical points, and both ways a point can be one, from the previous topic
• Local maxima and local minima as Definition 4 states them
• The sign of the derivative and what it reports about rising and falling
• Factorising a derivative and reading the sign of a product
• Differentiating a square root and a quotient, from Chapter 5
• One-sided behaviour of a function defined by cases

What they should be able to do

• State all three parts of Theorem 3 and say which hypothesis each needs
• Explain why the test asks only for continuity at the critical point, not differentiability
• Classify a critical point by testing the sign of the derivative on each side
• Build a table of nearby signs and read the verdict off it
• Recognise the case where the sign does not change, and name it
• Apply the test where the derivative does not exist at the point, by examining the two sides separately
• Say why the first test is the only available one at a point of non-differentiability
• Handle a derivative with a repeated factor, and predict from the factor whether the sign changes there
• Distinguish a local maximum from the local maximum value, in the chapter's own wording
• Read the Summary's version of the test and say what it leaves out

Where it usually goes wrong

• "A critical point is either a maximum or a minimum." Part (iii) exists precisely for the third case, and both Example 18 and Miscellaneous Exercise Q10 land on it. Teach the test as a three-way question from the first sentence.
• "The derivative has to exist at the critical point." Theorem 3 asks only that the function be unbroken there. Example 19 runs the test at a point with no derivative, and Fig 6.14 draws two such points. This is the test's main advantage over the next one.
• "I should evaluate the derivative at the critical point to see the sign." It is zero there, or undefined. The sign is read on each side, at a convenient input strictly between this critical point and the next.
• "Any nearby number will do as a test input." Any number strictly between this critical point and the neighbouring one will. A number chosen beyond the next critical point reports the wrong sign, and with three critical points that mistake is easy.
• "A repeated factor still changes sign." A factor raised to an even power never does. Miscellaneous Exercise Q10's squared factor is what makes one of its three critical points a non-extreme, and a student who tests numerically without noticing will get it right by luck and be unable to explain it.
• "The point and the value are the same answer." They are two answers, and the chapter has a boxed note about it. Questions in Exercise 6.3 Q3 ask for both explicitly.
• "The two spellings must mean two different things." They do not. The chapter uses one spelling on Part I p. 164 and the other from Part I p. 166 onward, including in the Summary. Say so once and move on.
• "If the second test is available I should always use it." The chapter's own note after Example 23 says otherwise: it chose the first there because it was shorter. Choose by the algebra in front of you.

Questions to check understanding

• State Theorem 3's three parts and the hypothesis it places at the critical point
• Classify every critical point of a stated polynomial and give the value at each — the form of Exercise 6.3 Q3
• Classify a critical point at which the function has no derivative
• Produce a function with a critical point that is neither kind of extreme
• Predict from a factored derivative, without substituting, which of its zeros change sign — the form of Miscellaneous Exercise Q10
• Explain why the derivative is examined on either side of the point rather than at it
• Given a point of local maxima, state the local maximum value

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 3 (§6.4, Part I p. 164). Stated for a function on an open interval, unbroken at a critical point of that interval. Part (i): if the derivative runs positive on the left of the point and negative on its right, the point is a local maximum. Part (ii): the mirror image gives a local minimum. Part (iii): if the sign is the same on both sides, the point is neither, and the chapter names that case. Section 2 should read the hypothesis aloud — the function need only be unbroken at the point, and the derivative is examined near it, never at it. That is what lets the test reach the corners of Fig 6.14.
• The boxed note after Theorem 3 (Part I p. 165). One sentence separating a point of local maxima from the local maximum value taken there, and the same for minima. Small, and it prevents a whole family of answer-writing errors: the question usually asks for both and students hand back one.
• Fig 6.13 and Fig 6.14 as the explanation (Part I pp. 164–165). The chapter says outright that these two figures explain Theorem 3 geometrically. Fig 6.13 is the cubic with the flat crossing at the origin — part (iii). Fig 6.14 carries four critical points, the first two smooth and the last two drawn as sharp corners, each labelled with its verdict — parts (i) and (ii), twice each, once smoothly and once not. Read closely. Section 5 is the two figures and the sentence that ties them to the theorem.
• Example 17 and its table (Part I p. 165). A cubic whose derivative factors into three times two linear factors, vanishing at one and at minus one. Verified: to the right of one the derivative is positive and to the left it is negative, so one is a local minimum with value one; to the left of minus one the derivative is positive and to the right negative, so minus one is a local maximum with value five. The chapter prints a table with a row for each critical point split into a right side and a left side, and a column carrying the sign, with sample inputs given in brackets. Sections 6 and 7 are this example and this table.
• The sample inputs in that table (Part I p. 165). The chapter offers concrete numbers just to one side of each critical point — a tenth above and a tenth below — and marks them as illustrative. This is the practical technique of the whole test and the chapter never states it as a rule. Section 7 should promote it: pick any input strictly between this critical point and the next, evaluate the factored derivative, and read the sign. It works because the derivative is continuous and has no zero in between.
• Example 18 (Part I p. 166). A cubic whose derivative is six times a perfect square, vanishing only at one. Verified: a square is non-negative, so the derivative is positive on both sides of one and zero only at it; the sign does not change, and by part (iii) the point is neither kind of extreme. The chapter names the verdict, using its second spelling. Section 8. A Remark follows observing that a derivative which never changes sign leaves the graph with no turning points at all — and names this example by a number the chapter does not have; see Notes.
• Example 19 and Fig 6.15 (Part I pp. 166–167). Three plus the modulus. The chapter observes there is no derivative at zero, says the second test is therefore unavailable, and runs the first: to the left the function is three less the input with derivative minus one, to the right three plus the input with derivative one. Verified: negative then positive, so zero is a local minimum with value three. Fig 6.15 is a V with its vertex lifted off the axis, its two arms labelled with their separate rules and the horizontal axis ticked from minus three to three. Section 9, and section 10's first half.
• Example 23 and the Note after it (Part I pp. 168–169). The shortest distance from a point on the vertical axis to a parabola, with the height of that point confined to a stated range. The chapter parametrises by the height of the parabola point, differentiates the distance, finds one critical value, and settles it by the sign on each side. Verified: the critical height is half of one less than twice the given height, the derivative is negative below it and positive above, so it is a minimum, and substituting gives the distance as the square root of one less than four times the given height, all halved. A boxed note then says the first test was preferred here because it is shorter. Section 10's second half is that note: the two tests are alternatives, and the chapter itself sometimes picks the first.
• Exercise 6.3 Q3 (Part I p. 175). Eight functions to be classified, with the values asked for as well as the points. Verified: (i) the square has a local minimum at zero with value zero. (ii) a cubic less three times the input has critical points at one and minus one, a local minimum of minus two and a local maximum of two. (iii) sine plus cosine on a quarter turn turns at an eighth of a turn, a local maximum of the square root of two. (iv) sine less cosine on a full turn turns at three eighths and at seven eighths, giving a local maximum of the square root of two and a local minimum of its negative. (v) a cubic with critical points at one and three, a local maximum of nineteen and a local minimum of fifteen. (vi) half the input plus two over it, on the positive reals, turns at two with a local minimum of two. (vii) the reciprocal of two more than the square turns at zero with a local maximum of a half. (viii) the input times the square root of one less it, on the unit interval, turns at two thirds with a local maximum of twice the square root of three, over nine.
• Miscellaneous Exercise Q10 (Part I p. 184). A product of a fourth power and a cube, with all three verdicts asked for by name. Verified: the derivative factors as the cube of two less than the input, times the square of one more than it, times seven times the input less two. The squared factor never changes sign, so at minus one the derivative vanishes without changing sign — the third verdict. At two sevenths the last factor turns, and the product goes from positive to negative — a local maximum. At two the cubed factor turns, and the product goes from negative to positive — a local minimum. Section 11 is this item; it is the only place in the chapter where a student must predict a verdict from the parity of an exponent rather than by substituting, and it is the best question in the exercise.
• The Summary's First Derivative Test bullet (Part I pp. 185–186). All three parts restated faithfully, including the third and its name. Verified against Part I p. 164 line by line. What it drops is the boxed note distinguishing a point from its value, and the sample-input technique. Section 12.

Figures to have open

• A critical point drawn with the derivative's sign unknown on both sides, for section 1. Not in the book.
• A three-row card of Theorem 3's parts for section 2. Content is the chapter's own Part I p. 164; layout is added here.
• Redraws of Fig 6.13 and Fig 6.14 (Part I pp. 164–165), with all four of Fig 6.14's critical points and its two sharp corners kept as corners. The chapter's own. Fig 6.14 is shared with the previous topic and should be the same drawing in both videos.
• A sign table for Example 17 with the sample inputs visible, for section 7. The content is the chapter's own from Part I p. 165. Build it with the repo's DataTable component.
• A redraw of Fig 6.15 (Part I p. 166): a V with its vertex lifted above the axis, each arm labelled with its own rule. The chapter's own.
• A number line with three critical points and a badly chosen test input beyond the next one, for the misconception in section 7. Not in the book.

Where this sits in the book

• NCERT Class 12 Mathematics, Part I, Chapter 6 "Application of Derivatives", §6.4, Theorem 3, Part I p. 164
• The boxed note on points against values, and the sentence tying Fig 6.13 and Fig 6.14 to the theorem, Part I p. 165
• Example 17 and its table, Part I p. 165; Example 18 and its Remark, Part I p. 166
• Example 19 with Fig 6.15, Part I pp. 166–167
• Example 23 and the boxed note following it, Part I pp. 168–169
• Exercise 6.3, question 3, Part I p. 175; Miscellaneous Exercise on Chapter 6, question 10, Part I p. 184
• Summary, the First Derivative Test bullet, Part I pp. 185–186