Local maxima and minima, and why critical points include the non-differentiable ones

What to assume they know

• Increasing and decreasing on an interval, from the previous module
• The sign of the derivative and what it reports, from the previous module
• Differentiability at a point, and functions that fail it, from Chapter 5
• Graphs of the square, the cube, the modulus and a simple linear function
• Interval notation, open and closed, and what an open interval excludes
• Quantifiers: a claim about every element of a set, and a claim about some element

What they should be able to do

• Read the three motivating problems and say which single mathematical question all three ask
• State Definition 3 as it is printed and explain why the comparison sign it uses makes the definition unsatisfiable
• Repair the definition and say what the repaired version asserts
• Decide, for a stated function on a stated interval, whether a highest value exists, a lowest value exists, both or neither
• Explain why a function on an open interval can fail to attain either, and give the chapter's own argument for it
• Exhibit a lowest value at a point where the function has no derivative
• Distinguish a value that is highest across the whole interval from one that is highest only nearby
• State Definition 4 and name the two ways it differs from Definition 3
• Argue from the sign of the derivative on either side that a smooth local extreme forces the derivative to vanish
• Give a function whose derivative vanishes at a point that is neither a local maximum nor a local minimum
• Define a critical point and list both ways a point can be one

Where it usually goes wrong

• "A maximum has to be at a turning point." Fig 6.7 panel (b) puts one at an end of the interval, and Example 14's note puts one at the left end of a restricted domain. Turning points are one source of extremes and not the only one.
• "The derivative is zero at every maximum." Only at smooth ones. Example 15's lowest value sits where the modulus has no derivative at all, and Fig 6.14 draws two such points out of four. Theorem 2 is the statement that covers both cases.
• "If the derivative is zero the point is a maximum or a minimum." The cube refutes it at the origin, and the chapter says so in a Remark with its own figure. This is the single most common error in the whole of §6.4.
• "Local and global are the same when there is only one turning point." They can still differ, because an end of the interval can beat the turn. Keep the two words separate from the moment Fig 6.11 appears.
• "Every function on an interval has a highest and a lowest value." Not on an open one. Example 16 is a rising line on an open interval with neither, and its argument — that no point is closest to the end — is the one to reproduce.
• "A local maximum must be strictly higher than its neighbours." Definition 4 allows equality. A function that is flat across a stretch has a local maximum at every point of the stretch, which is odd but correct under the printed definition.
• "The definition compares the point with the whole domain." Definition 4 compares it only within some width around it, and the width may be as small as you like. That is the entire content of the word local.
• "A sharp corner is a mistake in the drawing." It is the second case of Theorem 2, drawn deliberately, twice, in Fig 6.14. Students trained on smooth polynomials read those two marks as sloppiness.

Questions to check understanding

• State Definition 3 as printed and say what goes wrong when the chosen point is substituted into its own condition
• Decide for a stated function on a stated interval whether a highest value, a lowest value, both or neither exist — the form of Exercise 6.3 Q1 and Q2
• Show that a stated function has neither, by an argument about its derivative and its range — the form of Exercise 6.3 Q4
• Give a function with a local extreme at a point where it has no derivative
• Give a function whose derivative vanishes at a point that is not a local extreme
• State Theorem 2 and both cases it allows
• Define a critical point and mark all critical points on a supplied graph
• Explain why a function on an open interval may attain neither extreme

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.

• The §6.4 opening (Part I pp. 159–160). Two paragraphs and three numbered problems. The paragraphs announce turning points, the highest and lowest reaches of a graph locally, and — separately — the largest and smallest across a whole interval. The three problems are an orchard's profit as a quadratic in the number of trees per acre, a thrown ball whose path is a printed quadratic in horizontal distance, and a helicopter on a printed parabola with an observer at a stated point. Section 1 is these three and the sentence that follows them, which says all three ask one question. The third problem's observer is placed at one point here and at a different point in the worked example that solves it much later; see Notes.
• Definition 3 (Part I p. 160). Read closely in two separate close-ups, because the claim about it is a claim about a missing bar. Part (a) says the function has a highest value on the interval if there is a point of the interval whose value strictly exceeds the value at every point of the interval; part (b) is the mirror image with a strict comparison the other way; part (c) names either of them an extreme value and the point an extreme point. Both comparisons are printed strict, with no bar. Section 2 states it exactly as printed and section 3 dismantles it.
• Why it fails (not in the book). Verified: part (a) quantifies over every point of the interval, and the chosen point is in the interval, so the requirement applied at that point demands the value strictly exceed itself. Nothing satisfies it. Under Definition 3 as printed, no function has a highest value on any interval. The repair is the non-strict comparison, and the chapter's own Example 14 uses the non-strict form four lines later. Section 3 is the whole argument and it takes ninety seconds.
• Fig 6.7, panels (a), (b) and (c) (Part I p. 161). Three small axis pairs drawn above the first example. Panel (a) is a downward arch with its peak above a marked point on the horizontal axis and the height at that point drawn as a dashed segment and labelled. Panel (b) is a short rising segment over the interval from one to four, with dashed verticals at both ends — the highest value sits at the right-hand end. Panel (c) is an upward valley with a marked point and its labelled height. The Remark beneath them says these graphs help locate a highest or lowest value, including at a point where the function is not differentiable, and points forward to the modulus example. Section 4 is these three panels, and panel (b) is the one that earns its place: the extreme is at an end, not at a turn.
• Example 14 and Fig 6.8 (Part I p. 161). The square function on the whole line. Verified: the value is zero at zero and non-negative everywhere, so zero is the lowest value and it occurs at zero; there is no highest value, because the square grows without bound. The chapter's argument uses a non-strict comparison, which is the repaired Definition 3 and not the printed one. Fig 6.8 is an upward parabola through the origin with the horizontal axis ticked from minus three to three. A boxed note then observes that restricting the domain to a stated closed interval produces a highest value at the left-hand end — the first appearance in the chapter of the idea that ends can be extremes.
• Example 15 and Fig 6.9 (Part I pp. 161–162). The modulus on the whole line. Verified: the value is non-negative everywhere and zero at zero, so zero is the lowest value; there is no highest value. Fig 6.9 is a V through the origin with the vertical axis ticked at two and three. Two boxed notes follow: one restricting the domain again, and one observing that the function has no derivative at zero. The second note names the example by a number this chapter does not have; see Notes. Section 6 is this example, and its point is that the lowest value sits exactly where the derivative fails to exist.
• Example 16, Fig 6.10 and the Remark (Part I p. 162). The identity function on the open interval from zero to one. The chapter argues that no point is closest to either end — halving reaches a smaller input, and averaging with one reaches a larger — so neither a highest nor a lowest value exists. Fig 6.10 is a rising segment drawn between two open circles with a dashed vertical at the right end. The Remark then closes the interval and both values appear, and states two results it says are beyond the book's scope: a function that only rises or only falls attains its extremes at the ends of its domain, and a interval with both ends included, an unbroken function has both. A boxed note defines what "only rises or only falls" means. Section 7 is this example; sections 4 and 7 together set up the closed-interval topic later in this module.
• Fig 6.11 and the turning-point paragraph (Part I pp. 162–163). A wave with four marked points on it, two at the bottom of valleys and two at the top of hills, with dashed verticals dropping to the axis and the words for rising and falling written along the arcs. Read on the printed page. The paragraph beneath names the two valley points local minima and the two hill points local maxima, and gives the alternative names. Section 8 is this figure, and the move it makes is the topic's second big one: from "across the whole interval" to "compared with its immediate surroundings".
• Definition 4 (Part I p. 163). Read closely. Part (a) makes a point a local maximum if there is a positive width such that the value at the point is at least the value at every input within that width, excluding the point itself; part (b) is the mirror with at most, and does not carry the exclusion. Both comparisons carry the bar. Two things to say in section 9: the comparison is non-strict here where Definition 3 printed it strict, and the exclusion clause appears on one part and not on the other. Neither asymmetry changes any answer; both are worth one sentence.
• Fig 6.12, panels (a) and (b) (Part I p. 163). Two small axis pairs. Panel (a) is a smooth hill with an arrow to the peak labelled with the vanishing derivative, the left arc labelled with a positive derivative and the right arc with a negative one, and the input marked below. Panel (b) is the smooth valley with the two signs swapped. The paragraph between them runs the argument: rising before, falling after, so the derivative must be zero at the point. Sections 10 uses this pair.
• Theorem 2 (Part I p. 164). Stated without proof: at a local extreme on an open interval, either the derivative is zero or the function is not differentiable there. This is the theorem that forces section 12's second case, and it is the only place before Fig 6.14 that the non-differentiable possibility is named.
• The Remark and Fig 6.13 (Part I p. 164). The converse fails: the cube has a vanishing derivative at zero and no local extreme there. Fig 6.13 is the cubic through the origin, labelled with its rule, with a dot at the origin and the words for the flat-crossing point set beside it on the axis. Verified: the derivative is three times the square, zero only at zero, and positive on both sides, so the function rises through the point. Section 11.
• The boxed note defining a critical point (Part I p. 164). Two ways to qualify: the derivative is zero there, or the function is not differentiable there. The note adds a second sentence about a differentiable neighbourhood when the function is continuous and the derivative vanishes. Section 12 opens here.
• Fig 6.14 (Part I p. 165). Read closely, because everything asserted about it is a claim about what is drawn. One curve carrying four marked inputs on the horizontal axis. The first is a smooth hill with the vanishing derivative labelled and the word for a local maximum above it. The second is a smooth valley with its vanishing derivative labelled and the word for a local minimum beside it. The third is a sharp peak and the fourth a sharp valley, each labelled as a point of non-differentiability and as a local extreme of the corresponding kind. Dashed verticals drop from all four to the axis, and the arcs between them carry the signs of the derivative. This is the closing figure of the topic and the single most important picture in §6.4.
• Exercise 6.3 Q1 (Part I p. 174) and Q2 (Part I p. 175). Nine functions to be classified for highest and lowest values, with no derivative needed for most. Verified: Q1(i) is three plus a square, so its lowest value is three and it has no highest; (ii) completes to minus two plus a square of a linear expression, so its lowest value is minus two; (iii) is ten less a square, so its highest value is ten and it has no lowest; (iv) is a cubic and has neither. Q2(i) has lowest value minus one and no highest; (ii) has highest value three and no lowest; (iii) oscillates between four and six, so both exist; (iv) is the modulus of a quantity that runs between two and four, so it runs between two and four and both exist; (v) is on an open interval and has neither. Q2(iv) is worth a beat: the modulus does nothing, because the quantity inside never goes negative.
• Exercise 6.3 Q4 (Part I p. 175). Three functions to be shown to have neither. Verified: the exponential and the logarithm are strictly rising on their whole domains and unbounded, and the cubic-plus-quadratic-plus-linear has derivative three times the square plus twice the input plus one, whose discriminant is negative, so it is positive everywhere and the function is strictly rising and unbounded. This item is the bridge back to the previous module.

Figures to have open

• Three motivating panels for section 1. Not in the book; the chapter prints the three problems as text with no figure.
• A redraw of Fig 6.7 panels (a), (b) and (c) (Part I p. 161), with the dashed heights and the marked inputs. The chapter's own. Panel (b) must keep its two endpoint verticals, or section 4 loses its point.
• Redraws of Fig 6.8, Fig 6.9 and Fig 6.10 (Part I pp. 161–162): the upward parabola, the V, and the open segment between two open circles. The chapter's own. Fig 6.10's open circles are load-bearing for section 7.
• A redraw of Fig 6.11 (Part I p. 163): one wave with four marked points, two at the tops of hills and two at the bottoms of valleys, with dashed drops to the axis and the rising and falling arcs labelled. The chapter's own.
• A redraw of Fig 6.12 panels (a) and (b) (Part I p. 163), the smooth hill and the smooth valley with the signs of the derivative on each arc. The chapter's own.
• A redraw of Fig 6.13 (Part I p. 164): the cubic through the origin with the crossing point dotted. The chapter's own.
• A redraw of Fig 6.14 (Part I p. 165) with all four marked inputs, the first two smooth and the last two drawn as sharp corners, and every label kept. The chapter's own, and the topic's closing image.

Where this sits in the book

• NCERT Class 12 Mathematics, Part I, Chapter 6 "Application of Derivatives", §6.4 Maxima and Minima, the opening and the three problems, Part I pp. 159–160
• Definition 3, Part I p. 160; Fig 6.7 and the Remark, Part I pp. 160–161
• Examples 14, 15 and 16 with Fig 6.8, Fig 6.9 and Fig 6.10, and the boxed notes and the Remark, Part I pp. 161–162
• Fig 6.11 and the turning-point paragraph, Part I pp. 162–163
• Definition 4 and Fig 6.12, Part I p. 163
• Theorem 2, the Remark with Fig 6.13, and the boxed note on critical points, Part I p. 164; Fig 6.14, Part I p. 165
• Exercise 6.3, questions 1, 2 and 4, Part I pp. 174–175
• Summary, the critical point bullet, Part I p. 185