The quicker second-derivative test, and the case where it tells you nothing

What to assume they know

• The first derivative test, and the three verdicts it returns
• Critical points, and both ways a point can be one
• Second order derivatives, from Chapter 5
• Differentiating a quotient and a square root
• Reading the sign of a number obtained by substitution
• Similar triangles and the ratio of corresponding sides, for the cone example

What they should be able to do

• State Theorem 4's three parts with the correct sign in each
• Explain why a negative second derivative at a critical point corresponds to a local maximum, using the first test rather than memory
• State the hypothesis Theorem 4 places at the point, and name what it rules out
• Identify both situations in which the test returns nothing, and give a function exhibiting each
• Classify several critical points of one function by evaluating the second derivative at each
• Fall back to the first test when the second returns nothing, and complete the classification
• Use the test inside an optimisation where the second derivative is laborious to compute
• Recognise when the second derivative is constant and the test collapses to a single sign check
• Choose between the two tests on the basis of the algebra in front of you
• Read the Summary's version of the test and identify what it changes

Where it usually goes wrong

• "The second test replaces the first." It cannot: it needs two derivatives where the first needs none at the point, and it returns nothing when the second derivative vanishes. Both gaps are met inside this chapter, in Examples 19 and 21.
• "A positive second derivative means a maximum." It means a minimum. The argument in section 3 is the cure — a positive second derivative means the first derivative is climbing through zero, so the function falls then rises. Memorised signs invert; derived signs do not.
• "If the test fails, the point is a point of inflexion." Failure means the test says nothing. Part (iii) sends the reader back to the first test, which may return any of the three verdicts. Example 21 happens to land on the third one, which makes this misconception easy to acquire from a single example.
• "The second test is always quicker." Example 25's second derivative runs half a page. Look at the first derivative before choosing: if it is a polynomial, differentiate again; if it is a quotient with a root in it, test the sign on each side instead.
• "I only need the second derivative at one critical point." Example 20 has three, and each needs its own substitution. A single evaluation classifies a single point.
• "If both derivatives vanish, something has gone wrong." Nothing has. It is the ordinary situation for a repeated factor, and it is why the chapter prints a third part rather than two.
• "The test needs the function to be twice differentiable everywhere." Only at the point. Theorem 4 says so, and the boxed note beneath it says it again.
• "The Summary's unqualified maxima and minima mean what the theorem's qualified pair means." In this chapter they do not: the unqualified pair is Definition 3's whole-interval notion and the qualified pair is Definition 4's. The Summary's third part blurs them.

Questions to check understanding

• State Theorem 4's three parts with the correct sign in each
• Explain, without quoting the theorem, why a negative second derivative at a critical point gives a local maximum
• Classify every critical point of a stated polynomial using the second test
• Identify a critical point at which the second test returns nothing, and finish the classification another way
• Give a function whose local extreme cannot be classified by the second test at all, and say why
• Choose the more economical of the two tests for a stated function, and justify the choice
• Solve a stated optimisation and confirm the verdict with the second derivative

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 4 (§6.4, Part I p. 166). Stated for a function on an interval, twice differentiable at a point of it. Part (i): a vanishing first derivative and a negative second derivative make the point a local maximum, and the value there the local maximum value. Part (ii): a vanishing first derivative and a positive second derivative make it a local minimum. Part (iii): if both vanish the test returns nothing, and the reader is sent back to the first test to decide among the three verdicts. Read closely. Both signs in parts (i) and (ii) are strict as printed, which is correct and worth confirming aloud, because the two definitions on either side of this theorem in the chapter are non-strict.
• Why the signs go that way (not in the book). Verified: the second derivative is the rate at which the first one changes. If it is negative at a critical point then the first derivative is falling there, and since it is zero at the point it must be positive just before and negative just after — which is precisely part (i) of the first test, so the point is a local maximum. Section 3 is this argument. The chapter states Theorem 4 without proof and offers no reason for the signs, and students who memorise them invert them under exam pressure roughly half the time.
• The boxed note after Theorem 4 (Part I p. 166). One sentence spelling out what the hypothesis means: the second order derivative exists at the point. Section 4 should say what that excludes — a corner, where the first derivative does not exist, and also a point where the first derivative exists but the second does not.
• Example 19's opening line (Part I p. 166). Before running the first test on three plus the modulus, the chapter observes that the function has no derivative at zero and therefore that the second test is unavailable. Verified: with no first derivative at the point there is no second one either, so Theorem 4's hypothesis fails outright. Section 5 is this one line, and it is the chapter's own statement of the first failure mode.
• Example 20 (Part I p. 167). A quartic whose first derivative factors into twelve times the input, times one less than it, times two more than it, so the critical points are zero, one and minus two. The second derivative is twelve times a quadratic. Verified: at zero it is minus twenty-four, so zero is a local maximum with value twelve; at one it is thirty-six, so one is a local minimum with value seven; at minus two it is seventy-two, so minus two is a local minimum with value minus twenty. Three critical points, three substitutions, three verdicts — this is section 6 and it is the strongest argument for the test. The printed sentence reporting the two minima names the wrong input for one of them; see Notes.
• Example 21 (Part I p. 167). The cubic from Example 18 again. Its first derivative is six times a perfect square, vanishing at one, and its second derivative is twelve times one less than the input, which is also zero at one. Verified: both derivatives vanish, so part (iii) applies and the test returns nothing. The chapter then says it will go back to the first test, and quotes Example 18's verdict rather than redoing the work. Sections 7 and 8 are this example, and the pairing with Example 18 is the whole lesson: the same point, two tests, one answer, and only one of the tests got there.
• Example 22 and its Remark (Part I pp. 167–168). Two positive numbers with a fixed sum, minimising the sum of their squares. Verified: writing one number as the sum less the other gives twice the square of the input, less thirty times it, plus two hundred twenty-five; the first derivative vanishes at seven and a half, and the second derivative is the constant four, positive, so the point is a minimum and both numbers are seven and a half. The Remark generalises to an arbitrary fixed sum and reports that the two numbers are each half of it — and names the example by a number the chapter does not have; see Notes. Section 10 is this example: when the function is a quadratic the second derivative is a constant, and the test is a single glance.
• Example 24 and Fig 6.16 (Part I p. 169). Two vertical poles of stated heights at the ends of a stated horizontal span, with a point on the span chosen to minimise the sum of the squared distances to the two tops. Verified: with the distance from one end written as the variable, the sum is twice the square of the input, less forty times it, plus one thousand one hundred forty; the first derivative vanishes at ten and the second derivative is the constant four, so ten is the minimum — the midpoint of the span, which the chapter does not remark on. Fig 6.16 shows the two poles upright at the ends, the span marked in two pieces, and the two slant lines meeting at the chosen point. A second instance of section 10.
• Example 25 (Part I pp. 169–170). A trapezium with three sides of stated equal length, maximising the area. The chapter reaches an area that is a product of a linear factor and a square root, differentiates, finds the positive critical value, and then computes a second derivative that takes half a page before evaluating it as a negative number. Verified: the critical value is five, and the second derivative there is negative, so the area is greatest there and equals seventy-five times the square root of three. Section 9 is this example, and its point is honest: the second test is quicker in principle and can be far slower in practice, and the chapter does not warn the reader before setting off.
• Example 26 and Fig 6.18 (Part I pp. 170–171). A cylinder inscribed in a cone, maximising the curved surface. The chapter uses similar triangles to write the cylinder's height in terms of its radius, forms the surface as a constant times a quadratic in that radius, and differentiates twice. Verified: the first derivative vanishes when the cylinder's radius is half the cone's, and the second derivative is a negative constant, so that is the maximum. Fig 6.18 is a cone with its apex at the top and its axis drawn as a dashed vertical, with the inscribed cylinder drawn inside it and both circles shown as ellipses. A third instance of section 10, and the cleanest.
• Exercise 6.3 Q3 (Part I p. 175). The same eight functions the first-test topic uses. Six of the eight can be settled by the second test in one substitution each; two of them — the one on a bounded interval and the one with a square root — are quicker by the first. Section 11 should walk one of each. Verified for (ii): the second derivative is six times the input, so it is positive at one and negative at minus one, giving a local minimum of minus two and a local maximum of two.
• The Summary's Second Derivative Test bullet (Part I p. 186). All three parts restated. Verified against Part I p. 166 word by word: the third part is changed — where the theorem sends the reader back to decide among local maxima, local minima and the third verdict, the Summary drops the qualifier and offers maxima, minima and the third verdict. Section 12 is this comparison. In this chapter the unqualified words carry a different meaning, reserved for the whole-interval notion, so the Summary's version is not a harmless abbreviation.

Figures to have open

• A side-by-side of a two-sided sign check and a single substitution, for section 1. Not in the book.
• A three-row card of Theorem 4's parts for section 2, with the signs prominent. Content is the chapter's own Part I p. 166; layout is added here.
• A graph of the first derivative crossing zero downwards, with the function above it, for section 3. Not in the book, and the load-bearing picture of the topic; the chapter offers no reason for the signs and prints no figure with Theorem 4.
• A redraw of Fig 6.16 (Part I p. 169): two upright poles of unequal height at the ends of a horizontal segment, a marked point between them, and the two slant lines to the tops. The chapter's own.
• A redraw of Fig 6.18 (Part I p. 171): a cone with the apex at the top, the axis dashed, and an inscribed cylinder drawn with both its circles as ellipses. The chapter's own.
• A three-column table for section 10 of the problems whose second derivatives are constants. An added layout over the chapter's own examples. Build it with the repo's DataTable component.

Where this sits in the book

• NCERT Class 12 Mathematics, Part I, Chapter 6 "Application of Derivatives", §6.4, Theorem 4 and the boxed note beneath it, Part I p. 166
• Example 19's opening observation, Part I p. 166
• Examples 20 and 21, Part I p. 167; Example 22 and its Remark, Part I pp. 167–168
• Example 24 with Fig 6.16, Part I p. 169; Example 25, Part I pp. 169–170; Example 26 with Fig 6.18, Part I pp. 170–171
• Exercise 6.3, question 3, Part I p. 175
• Summary, the Second Derivative Test bullet, Part I p. 186