Applications of derivatives: synthesis and challenges

Summary

This lesson closes the unit on the applications of derivatives and, with it, the course. It gathers into a single deductive chain what the nine lessons of the unit proved: the Weierstrass theorem, obtained from the completeness axiom like Bolzano’s theorem, and Fermat’s theorem, obtained from the sign of the derivative at a point, yield Rolle’s theorem; from it come the mean value theorems of Lagrange and of Cauchy, and from these L’Hôpital’s rule, monotonicity from the sign of the derivative, the tests for extrema, convexity, optimization and the error bound of the Newton-Raphson method. It explains how the unit rests on the two preceding ones and on Algebra and Geometry, and what it leaves to the course that follows: the integral, sequences and series, and the Taylor polynomial. It ends with an exit assessment, parallel to the entry assessment, and with ten challenges, harder than the problems of the lessons, each of which combines results from several of them and from the earlier units.

Learning objectives

Set out in logical order the main theorems of the unit, indicating which earlier results each one depends on and in which proofs the Weierstrass theorem, the mean value theorem and monotonicity from the sign of the derivative are reused.
Distinguish what the unit admits from what it proves, recognize the hypotheses of each result and point out where the completeness axiom intervenes.
Relate the results of the unit to the units on limits and on derivatives, to Algebra and Geometry and to what Calculus II studies.
Solve nonroutine problems that combine the existence of extrema, the mean value theorem, L’Hôpital’s rule, monotonicity, convexity, curvature, optimization and the Newton-Raphson method.

What the unit achieved

The two preceding units worked at a point: the limit at a point, continuity at a point, the derivative at a point. This unit moved from the pointwise to the global, and the bridge was the completeness axiom. The Weierstrass theorem, proved with the supremum like Bolzano’s theorem, ensures that a function continuous on a closed and bounded interval attains its maximum and its minimum; Fermat’s theorem, which follows from the sign of the derivative at a point, tells where the interior extrema can be; and together they yield Rolle’s theorem, from which the two mean value theorems follow. Lagrange’s is written as follows:

f(b) - f(a) = f'(c)(b - a), \qquad c \in \ioo{a}{b}

That equality turns information about the derivative at all the points of an interval into information about the values of the function, and almost everything else in the unit is a reading of it: monotonicity, the tests for extrema, convexity, L’Hôpital’s rule, through Cauchy’s version, and the error bound of the Newton-Raphson method. The proofs cite these theorems instead of redoing the argument.

Existence and the mean value

The lesson on the Weierstrass theorem first proved the boundedness theorem, taking the supremum of the points up to which the function is bounded and moving beyond it with the local boundedness that continuity provides; then, that the supremum of the values is attained, because if it were not attained, the reciprocal of the distance from the values to it would be continuous and unbounded. It showed with counterexamples that no hypothesis is superfluous (an open interval, an unbounded one, a discontinuous function and the rationals, where completeness fails), deduced with the intermediate value theorem that the image of a closed and bounded interval is another interval of the same kind, and that a function with equal values at the endpoints of an interval attains an extremum in its interior. The lesson on the theorems of Fermat, Rolle, Lagrange and Cauchy defined local extrema as quantified statements, proved with the sign of the derivative at a point that the derivative vanishes at an interior local extremum, and derived in succession Rolle’s theorem, Lagrange’s mean value theorem and Cauchy’s mean value theorem, showing that their hypotheses are necessary. Its corollaries were the first global conclusions: a function with zero derivative on an interval is constant, two functions with the same derivative differ by a constant, a bound on the derivative bounds the increments, between two roots of a function there is a root of its derivative, and the limit of the derivative, if it exists, is the derivative. The lesson on L’Hôpital’s rule proved it with Cauchy’s theorem, for the form $\frac{0}{0}$ and, with more care, for the form $\frac{\infty}{\infty}$ , on one side of a point, at the point and at infinity; it reduced to them the forms $0 \cdot \infty$ , $\infty - \infty$ , $1^\infty$ , $0^0$ and $\infty^0$ , showed that the rule is inconclusive when the quotient of the derivatives has no limit, warned that it cannot be used for the limits with which the derivatives were computed, and obtained from it the corollary on vanishing derivatives at a point, which compares the increment of a function with a power of $x - a$ .

Monotonicity, extrema and the shape of the graph

The lesson on monotonicity proved, with the mean value theorem, that a nonnegative derivative on an interval gives an increasing function, and a positive one, a strictly increasing function, and characterized strict monotonicity with a nonnegative derivative: the derivative must not vanish on a whole interval. Thus the promises of the unit on derivatives were kept, such as the promise that a positive derivative on an interval suffices for the inverse to exist and be differentiable. It defined critical points, proved the first derivative test and Darboux’s theorem, according to which a derivative takes every intermediate value even if it is not continuous, so that it has no jump discontinuities, and deduced that a derivative without zeros on an interval does not change sign. The lesson on maxima and minima defined absolute and strict extrema; it proved the second derivative test and, with L’Hôpital’s rule and without the Taylor polynomial, the higher-order derivative test, which decides by the parity of the first order with a nonzero derivative; and it looked for the absolute extrema on a closed and bounded interval among the endpoints of the interval and the critical points, and on an open or unbounded interval with the limits at the endpoints and with monotonicity, which also shows that a local extremum at the only critical point is absolute. The lesson on concavity and convexity defined convexity by the position of the graph with respect to its chords, with the convention of the unit (convex, below its chords), and proved that, for a differentiable function, being convex is equivalent to having an increasing derivative and to the graph lying above its tangents, and for a twice differentiable one, to the second derivative not being negative; it defined inflection points, decided them by the change of sign of the second derivative or by the third derivative, showed with $x^4$ that a zero second derivative is not enough, and obtained Jensen’s inequality and the inequality between the geometric and arithmetic means. The lesson on curvature defined the osculating circle as the one that has at the point the same value, the same derivative and the same second derivative as the function; it proved with implicit differentiation that it exists and is unique when the second derivative does not vanish, with radius $\rho = \frac{\left(1 + f'(a)^2\right)^{\frac{3}{2}}}{\abs{f''(a)}}$ and center on the normal, on the side toward which the graph bends, and showed that this center is the limit of the intersections of nearby normals.

Optimization and approximation

The lesson on optimization problems set out a method in five stages (the variable, the function, the domain, the absolute extrema with their justification, and the conclusion), whose decisive point is to justify that the extremum found is absolute: with the Weierstrass theorem if the domain is closed and bounded, and otherwise with the sign of the derivative, convexity or the limits at the endpoints. It proved that a function and its composition with a strictly increasing function attain their extrema at the same points, that the point of a graph nearest to a given point exists and lies on a normal, and it deduced from least time the laws of refraction and of reflection. The lesson on the Newton-Raphson method replaced the graph by its tangent to approximate a root; it treated the iterates as a function defined on the natural numbers, with the definition of the limit at infinity and without the theory of sequences; it proved with convexity and the infimum of the iterates that, from the right side, they approach the root without passing it and converge to it, and that from the wrong side the first iteration jumps to the right side; it bounded the error with the mean value theorem, showed the failures of the method (a zero derivative, a cycle, iterates that escape) and compared it with bisection, which gains one binary digit at each iteration, whereas the Newton-Raphson method, near a simple root, doubles in the examples the number of correct digits.

What is admitted, what is proved and what remains

The balance of the unit can be summarized as follows:

The following are admitted: the completeness axiom, which the unit uses directly in the boundedness theorem, in the Weierstrass theorem and in the convergence of the Newton-Raphson iterates; the theorems of the units on limits and on derivatives, in particular Bolzano’s theorem, the preservation of sign, the local character of the derivative and the sign of the derivative at a point; and results of Algebra and Geometry, such as the trigonometric formulas, the inequality $\sin x < x < \tan x$ and the laws of exponents and of logarithms.
The unit proves facts that the unit on derivatives had announced without being able to prove them: that a positive derivative on an interval gives an increasing function, that a function with zero derivative on an interval is constant and that the error of the approximation by the differential can be bounded; and the tangent becomes a tool for approximating roots, for comparing the graph with itself and for measuring how much it bends.
Three tools are reused instead of repeating arguments: the Weierstrass theorem, which yields the extrema; the mean value theorem, which translates the sign of the derivative into information about the values; and monotonicity from the sign of the derivative, which underpins the tests for extrema and convexity.
What requires more than the derivative is left for Calculus II: the integral, arc length, with which curvature is read as the rate at which the tangent turns, uniform continuity, sequences and series, and the Taylor polynomial; and for Calculus III, the extrema of functions of several variables and the curvature of parametric curves.

The following figure gathers the three segments into a single chain. Each node of the column is a definition or a theorem, numbered according to the logical order and joined by an arc to the earlier links that its proof uses; around it float the supports, that is, the principles and tools to which the chain resorts again and again, each one joined by a solid line to the links that rest on it and by a dotted one to the link where it is established, except those that come from the earlier units or from Algebra and Geometry, which are drawn as rings. Clicking a support highlights all the links that use it, and each link refers to the lesson in which it is proved.

Supports of the chain

ACompleteness axiom. It is admitted, from Algebra and Geometry I, that the reals form a complete ordered field: every nonempty set of reals that is bounded above has a supremum, and every nonempty set that is bounded below has an infimum. Admitted without proof in this unit.
BInheritance from unit 1. Results proved in the unit on limits and continuity: the algebra of limits; one-sided limits, limits at infinity and infinite limits; the preservation of inequalities; the continuity of the operations and of composition; local boundedness and the preservation of sign, for limits and for continuous functions; and the standard limits. Admitted without proof in this unit.
CInheritance from unit 2. Results proved in the unit on derivatives: the definition of the derivative and one-sided derivatives, that differentiable implies continuous, the local character of the derivative, the differentiation rules and the chain rule, the derivatives of the elementary functions, the derivative of the inverse function and implicit differentiation, and higher-order derivatives. Admitted without proof in this unit.
DIntermediate value. Proved with the supremum in the unit on derivatives: if $f$ is continuous on $\icc{a}{b}$ and $f(a)$ and $f(b)$ have opposite signs, $f$ vanishes at some point of $\ioo{a}{b}$ , which is Bolzano’s theorem; and the image of an interval under a continuous function is an interval. Admitted without proof in this unit.
ESign of the derivative at a point. Proved in the unit on derivatives: if $f$ is defined on an open interval and $f'(a) > 0$ , there exists $\delta > 0$ such that $f(x) < f(a)$ if $a - \delta < x < a$ , and $f(x) > f(a)$ if $a < x < a + \delta$ ; it does not follow that $f$ is increasing near $a$ . Admitted without proof in this unit.
FExtrema attained. A function continuous on a closed and bounded interval attains its maximum and its minimum on it: the tool that gives existence to the extrema and to the intermediate points of the subsequent theorems. Established in: 2. Weierstrass theorem.
GMean value. If $f$ is continuous on $\icc{a}{b}$ and differentiable at each point of $\ioo{a}{b}$ , there exists $c$ in $\ioo{a}{b}$ with $f(b) - f(a) = f'(c)(b - a)$ : information about the derivative at all the points is turned into information about the values. Established in: 7. Lagrange’s mean value theorem.
HMonotonicity from the derivative. On an interval, a nonnegative derivative gives an increasing function, and a positive one, a strictly increasing function: the tool with which extrema and convexity are decided. Established in: 12. Monotonicity from the sign of the derivative.

Segment 1 Existence and the mean value
- Prove with the supremum that a function continuous on a closed and bounded interval is bounded and attains its maximum and its minimum, and show that no hypothesis is superfluous.
- Locate interior local extrema with Fermat’s theorem and deduce from it and from the Weierstrass theorem the theorems of Rolle, Lagrange and Cauchy.
- Obtain from the mean value theorem that a zero derivative gives a constant function, bound increments and count roots.
- Prove L’Hôpital’s rule with Cauchy’s theorem, reduce the other indeterminate forms to it and not use it for the limits with which the derivatives were computed.
1. Boundedness theorem. If $f$ is continuous on a closed and bounded interval $\icc{a}{b}$ , there exists a number $K$ such that $\abs{f(x)} \leq K$ for every $x$ in $\icc{a}{b}$ .
  Rests on: A (Completeness axiom), B (Inheritance from unit 1). Proved in: The Weierstrass extreme value theorem.
2. Weierstrass theorem. If $f$ is continuous on a closed and bounded interval $\icc{a}{b}$ , there exist $x_1$ and $x_2$ in $\icc{a}{b}$ such that $f(x_1) \leq f(x) \leq f(x_2)$ for every $x$ in $\icc{a}{b}$ ; none of the hypotheses can be omitted.
  Uses: 1. Boundedness theorem. Rests on: A (Completeness axiom), B (Inheritance from unit 1). Proved in: The Weierstrass extreme value theorem.
3. The image of a closed and bounded interval. If $f$ is continuous on $\icc{a}{b}$ , and $m$ and $M$ are its minimum and maximum values on it, the image of $\icc{a}{b}$ is the interval $\icc{m}{M}$ .
  Uses: 2. Weierstrass theorem. Rests on: F (Extrema attained), D (Intermediate value). Proved in: The Weierstrass extreme value theorem.
4. An extremum in the interior. If $f$ is continuous on $\icc{a}{b}$ , with $a < b$ and $f(a) = f(b)$ , it attains its maximum or its minimum at some point of $\ioo{a}{b}$ .
  Uses: 2. Weierstrass theorem. Rests on: F (Extrema attained). Proved in: The Weierstrass extreme value theorem.
5. Fermat’s theorem. If $f$ , defined on $A$ , attains a local extremum at an interior point $a$ of $A$ , that is, if there exists $\delta > 0$ such that $f(x) \leq f(a)$ for all $x$ in $A$ with $\abs{x - a} < \delta$ , or $f(a) \leq f(x)$ for all of them, and $f$ is differentiable at $a$ , then $f'(a) = 0$ .
  Rests on: E (Sign of the derivative at a point), C (Inheritance from unit 2). Proved in: The theorems of Fermat, Rolle, Lagrange and Cauchy.
6. Rolle’s theorem. If $f$ is continuous on $\icc{a}{b}$ , with $a < b$ , differentiable at each point of $\ioo{a}{b}$ and $f(a) = f(b)$ , there exists $c$ in $\ioo{a}{b}$ with $f'(c) = 0$ .
  Uses: 4. An extremum in the interior, 5. Fermat’s theorem. Rests on: F (Extrema attained), C (Inheritance from unit 2). Proved in: The theorems of Fermat, Rolle, Lagrange and Cauchy.
7. Lagrange’s mean value theorem. If $f$ is continuous on $\icc{a}{b}$ , with $a < b$ , and differentiable at each point of $\ioo{a}{b}$ , there exists $c$ in $\ioo{a}{b}$ with $f(b) - f(a) = f'(c)(b - a)$ : some tangent is parallel to the chord.
  Uses: 6. Rolle’s theorem. Rests on: B (Inheritance from unit 1), C (Inheritance from unit 2). Proved in: The theorems of Fermat, Rolle, Lagrange and Cauchy.
8. Cauchy’s mean value theorem. If $f$ and $g$ are continuous on $\icc{a}{b}$ , with $a < b$ , and differentiable at each point of $\ioo{a}{b}$ , there exists $c$ in $\ioo{a}{b}$ with $\left(f(b) - f(a)\right) g'(c) = \left(g(b) - g(a)\right) f'(c)$ ; if moreover $g'$ does not vanish on $\ioo{a}{b}$ , then $g(b) \neq g(a)$ and $\frac{f(b) - f(a)}{g(b) - g(a)} = \frac{f'(c)}{g'(c)}$ .
  Uses: 6. Rolle’s theorem. Rests on: B (Inheritance from unit 1), C (Inheritance from unit 2). Proved in: The theorems of Fermat, Rolle, Lagrange and Cauchy.
9. Consequences of the mean value theorem. Let $f$ be continuous on an interval $I$ and differentiable at each interior point. If $f'$ vanishes at them, $f$ is constant; two functions with the same derivative differ by a constant; if $\abs{f'(x)} \leq K$ at them, $\abs{f(x) - f(y)} \leq K \abs{x - y}$ for all $x$ , $y$ in $I$ ; and if $f$ is differentiable on $I$ and has $n \geq 2$ distinct roots in it, $f'$ has at least $n - 1$ .
  Uses: 6. Rolle’s theorem, 7. Lagrange’s mean value theorem. Rests on: G (Mean value), C (Inheritance from unit 2). Proved in: The theorems of Fermat, Rolle, Lagrange and Cauchy.
10. L’Hôpital’s rule. If $f$ and $g$ are differentiable on $\ioo{a}{b}$ , with $g'(x) \neq 0$ on it, if $f(x)$ and $g(x)$ tend to $0$ , or $g(x)$ tends to $\infty$ or to $-\infty$ , as $x \to a^+$ , and if $\frac{f'(x)}{g'(x)}$ tends to $L$ , real, $\infty$ or $-\infty$ , then $\frac{f(x)}{g(x)}$ tends to $L$ ; the same holds from the left, on both sides of a point and at infinity.
  Uses: 8. Cauchy’s mean value theorem. Rests on: B (Inheritance from unit 1), C (Inheritance from unit 2). Proved in: L’Hôpital’s rule.
11. Vanishing derivatives at a point. Let $n \geq 2$ be a natural number, $I$ an open interval, $a$ a point of $I$ and $f$ a function $n - 1$ times differentiable on $I$ and $n$ times differentiable at $a$ , with $f'(a) = \dots = f^{(n - 1)}(a) = 0$ . Then $\frac{f(x) - f(a)}{(x - a)^n}$ tends to $\frac{f^{(n)}(a)}{n!}$ as $x \to a$ .
  Uses: 10. L’Hôpital’s rule. Rests on: C (Inheritance from unit 2). Proved in: L’Hôpital’s rule.
Segment 2 Monotonicity, extrema and the shape of the graph
- Decide monotonicity by the sign of the derivative, characterize strict monotonicity and prove Darboux’s theorem.
- Classify critical points with the first derivative test, the second derivative test and the higher-order derivative test, and find absolute extrema on closed, open or unbounded intervals.
- Characterize convexity by chords, by the increasing derivative, by tangents and by the second derivative, and recognize inflection points.
- Measure the curvature of a graph with the osculating circle.
1. Monotonicity from the sign of the derivative. Let $f$ be continuous on an interval $I$ and differentiable at each interior point. If $f'(x) \geq 0$ at them, $f$ is increasing on $I$ , and if $f'(x) > 0$ , strictly increasing; with $f'(x) \leq 0$ or $f'(x) < 0$ , decreasing or strictly decreasing. Conversely, if $f$ is increasing on $I$ and differentiable at an interior point $a$ , then $f'(a) \geq 0$ .
  Uses: 7. Lagrange’s mean value theorem. Rests on: E (Sign of the derivative at a point), G (Mean value), C (Inheritance from unit 2). Proved in: Monotonicity and the first derivative test.
2. Strict monotonicity with a nonnegative derivative. If $f$ is continuous on an interval $I$ and $f'(x) \geq 0$ at each interior point, $f$ is strictly increasing on $I$ if and only if $f'$ does not vanish at all the points of any interval $\ioo{p}{q}$ contained in $I$ ; in particular, if it vanishes only at finitely many points.
  Uses: 9. Consequences of the mean value theorem, 12. Monotonicity from the sign of the derivative. Rests on: H (Monotonicity from the derivative), C (Inheritance from unit 2). Proved in: Monotonicity and the first derivative test.
3. First derivative test. If $f$ is continuous on $\ioo{a - r}{a + r}$ and differentiable at each point other than $a$ , with $f'(x) \geq 0$ to the left of $a$ and $f'(x) \leq 0$ to the right, it attains a local maximum at $a$ , strict if the inequalities are strict; with the inequalities reversed, a local minimum; and if $f'(x)$ is positive on both sides, or negative on both sides, there is no extremum at $a$ .
  Uses: 12. Monotonicity from the sign of the derivative. Rests on: H (Monotonicity from the derivative). Proved in: Monotonicity and the first derivative test.
4. Darboux’s theorem. If $f$ is differentiable on an open interval $I$ and $a < b$ are points of $I$ , $f'$ takes, on $\ioo{a}{b}$ , every value lying strictly between $f'(a)$ and $f'(b)$ ; consequently, $f'$ has on $I$ no removable or jump discontinuities.
  Uses: 2. Weierstrass theorem, 5. Fermat’s theorem. Rests on: E (Sign of the derivative at a point), F (Extrema attained), B (Inheritance from unit 1), C (Inheritance from unit 2). Proved in: Monotonicity and the first derivative test.
5. Derivative without zeros. If $f$ is continuous on an interval $I$ with more than one point and differentiable at each interior point, with $f'(x) \neq 0$ at each of them, $f'$ has the same sign at all of them and $f$ is strictly monotonic on $I$ .
  Uses: 12. Monotonicity from the sign of the derivative, 15. Darboux’s theorem. Rests on: H (Monotonicity from the derivative). Proved in: Monotonicity and the first derivative test.
6. Second derivative test. If $f$ is differentiable on an open interval $I$ and twice differentiable at a point $a$ of $I$ , with $f'(a) = 0$ , it attains at $a$ a strict local minimum if $f''(a) > 0$ , and a strict local maximum if $f''(a) < 0$ .
  Uses: 14. First derivative test. Rests on: E (Sign of the derivative at a point), C (Inheritance from unit 2). Proved in: Maxima and minima of a function.
7. Higher-order derivative test. If $f$ is $n - 1$ times differentiable on an open interval $I$ and $n$ times differentiable at a point $a$ of $I$ , with $n \geq 2$ , $f'(a) = \dots = f^{(n - 1)}(a) = 0$ and $f^{(n)}(a) \neq 0$ , then, if $n$ is even, $f$ attains at $a$ a strict local minimum if $f^{(n)}(a) > 0$ , and a strict local maximum if $f^{(n)}(a) < 0$ ; and if $n$ is odd, it attains no local extremum at $a$ .
  Uses: 11. Vanishing derivatives at a point. Rests on: B (Inheritance from unit 1). Proved in: Maxima and minima of a function.
8. Absolute extrema on an interval. If $f$ is continuous on $\icc{a}{b}$ , it attains its absolute maximum and minimum, and each point at which it attains them is $a$ , $b$ or a critical point in $\ioo{a}{b}$ ; if it is continuous on an open interval, bounded or not, and some value exceeds its limits at both endpoints, it attains its absolute maximum, at critical points.
  Uses: 2. Weierstrass theorem, 5. Fermat’s theorem. Rests on: F (Extrema attained), B (Inheritance from unit 1). Proved in: Maxima and minima of a function.
9. Absolute extrema by monotonicity. If $f$ is continuous on an interval $I$ , $c$ is an interior point, $f'(x) \geq 0$ at the interior points less than $c$ and $f'(x) \leq 0$ at those greater than it, $f$ attains at $c$ its absolute maximum on $I$ ; and if $c$ is the only critical point of $f$ in $I$ and $f$ attains a local extremum at it, that extremum is absolute and strict on $I$ .
  Uses: 12. Monotonicity from the sign of the derivative, 16. Derivative without zeros. Rests on: H (Monotonicity from the derivative). Proved in: Maxima and minima of a function.
10. Convexity and increasing derivative. A function $f$ is convex on an interval $I$ if $f\left(tx + (1 - t)y\right) \leq t f(x) + (1 - t) f(y)$ for all $x$ , $y$ in $I$ and every $t$ in $\icc{0}{1}$ . If $f$ is continuous on $I$ and differentiable at each interior point, it is convex if and only if $f'$ is increasing at the interior points, and strictly convex if and only if $f'$ is strictly increasing at them.
  Uses: 7. Lagrange’s mean value theorem. Rests on: G (Mean value), B (Inheritance from unit 1), C (Inheritance from unit 2). Proved in: Concavity, convexity and inflection points.
11. The graph above its tangents. If $f$ is continuous on an interval $I$ with more than one point and differentiable at each interior point, it is convex if and only if $f(x) \geq f(a) + f'(a)(x - a)$ for every interior point $a$ and every $x$ in $I$ ; in particular, if it is convex and $f'(a) = 0$ , it attains at $a$ its absolute minimum on $I$ .
  Uses: 21. Convexity and increasing derivative. Proved in: Concavity, convexity and inflection points.
12. Convexity and second derivative. If $f$ is continuous on an interval $I$ with more than one point and twice differentiable at each interior point, it is convex if and only if $f''(x) \geq 0$ at them, and it is strictly convex if moreover $f''$ does not vanish at all the points of any open interval contained in $I$ .
  Uses: 12. Monotonicity from the sign of the derivative, 13. Strict monotonicity with a nonnegative derivative, 21. Convexity and increasing derivative. Rests on: H (Monotonicity from the derivative), C (Inheritance from unit 2). Proved in: Concavity, convexity and inflection points.
13. Inflection points. If $f$ is continuous near $a$ , differentiable at $a$ and $f''$ changes sign at $a$ , the point $a$ is an inflection point; at an inflection point at which $f$ is twice differentiable, $f''(a) = 0$ , but that is not enough, as $x^4$ at $0$ shows; and if $f''(a) = 0$ and $f'''(a) \neq 0$ , $a$ is an inflection point and the graph crosses its tangent.
  Uses: 5. Fermat’s theorem, 18. Higher-order derivative test, 21. Convexity and increasing derivative, 23. Convexity and second derivative. Rests on: E (Sign of the derivative at a point), B (Inheritance from unit 1). Proved in: Concavity, convexity and inflection points.
14. Jensen’s inequality. If $f$ is continuous and convex on an interval $I$ and differentiable at each interior point, then $f\left(t_1 x_1 + \dots + t_n x_n\right) \leq t_1 f(x_1) + \dots + t_n f(x_n)$ for points $x_i$ of $I$ and weights $t_i \geq 0$ summing to $1$ ; with the logarithm, which is concave, the geometric mean of positive numbers does not exceed the arithmetic mean.
  Uses: 22. The graph above its tangents, 23. Convexity and second derivative. Rests on: C (Inheritance from unit 2). Proved in: Concavity, convexity and inflection points.
15. Osculating circle. If $f$ is differentiable on an open interval and twice differentiable at a point $a$ , with $m = f'(a)$ and $q = f''(a) \neq 0$ , there is exactly one circle that has at $a$ the same value, the same derivative and the same second derivative as $f$ ; its radius is $\rho = \frac{\left(1 + m^2\right)^{\frac{3}{2}}}{\abs{q}}$ , the reciprocal of the curvature $\kappa$ , and its center lies on the normal, above the point if $q > 0$ and below it if $q < 0$ .
  Rests on: C (Inheritance from unit 2). Proved in: Curvature and radius of curvature.
Segment 3 Optimization and approximation
- Translate a geometric or physical problem into the search for the absolute extremum of a function, on a precisely specified domain, and justify that the extremum found is absolute.
- Find the point of a graph nearest to a given point and deduce from least time the laws of reflection and refraction.
- Approximate roots with the Newton-Raphson method, ensure its convergence with convexity and bound the error with the mean value theorem.
1. The distance from a point to a graph. If $f$ is continuous on an interval with more than one point that contains its endpoints, if it has any, some point of its graph is at minimum distance from a point $P = (x_1, y_1)$ ; if the abscissa $c$ of that point is interior and $f$ is differentiable at it, $P$ lies on the normal: $(x_1 - c) + f'(c)\left(y_1 - f(c)\right) = 0$ .
  Uses: 2. Weierstrass theorem, 5. Fermat’s theorem. Rests on: F (Extrema attained), B (Inheritance from unit 1), C (Inheritance from unit 2). Proved in: Optimization problems.
2. Snell’s law. The duration $T(x) = \frac{\sqrt{a^2 + x^2}}{v_1} + \frac{\sqrt{b^2 + (d - x)^2}}{v_2}$ of the path that goes from $(0, a)$ to $(x, 0)$ at speed $v_1$ and from there to $(d, -b)$ at speed $v_2$ attains its strict absolute minimum at a single point, where $\frac{\sin \theta_1}{v_1} = \frac{\sin \theta_2}{v_2}$ ; hence, the shortest path between two points that touches a line makes equal angles with the perpendicular to it, which is the law of reflection.
  Uses: 22. The graph above its tangents, 23. Convexity and second derivative. Rests on: D (Intermediate value), C (Inheritance from unit 2). Proved in: Optimization problems.
3. Convergence of Newton-Raphson. If $f$ is differentiable and convex on an open interval $I$ , with $f'(x) \neq 0$ on all of $I$ , and $f(r) = 0$ , the iterates $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$ from an $x_0$ in $I$ with $f(x_0) \geq 0$ approach $r$ monotonically, without passing it, and converge to $r$ ; from an $x_0$ with $f(x_0) < 0$ , if $x_1$ is in $I$ , the first iteration leads to the right side.
  Uses: 16. Derivative without zeros, 21. Convexity and increasing derivative, 22. The graph above its tangents. Rests on: A (Completeness axiom), B (Inheritance from unit 1). Proved in: The Newton-Raphson method.
4. Error bound. If $f$ is continuous on an interval $J$ and differentiable at each interior point, with $\abs{f'(x)} \geq m > 0$ at them, and $f(r) = 0$ for some $r$ in $J$ , then $\abs{x - r} \leq \frac{\abs{f(x)}}{m}$ for every $x$ in $J$ .
  Uses: 7. Lagrange’s mean value theorem. Rests on: G (Mean value). Proved in: The Newton-Raphson method.

Toward Calculus II. The chain stops where the course ends. In Calculus II, the corollary according to which two functions with the same derivative differ by a constant is the foundation of antiderivatives, and the integral is joined to the derivative in the fundamental theorem of calculus; with arc length, which is an integral, curvature is read as the rate at which the tangent turns. Sequences, of which the Newton-Raphson iterates were a first example, and series give functions defined on the natural numbers a theory of their own. And the Taylor polynomial, whose remainder is bounded with the theorems of Rolle and Cauchy, extends the tangent line with the higher-order derivatives, generalizes the higher-order derivative test and proves, under hypotheses on the second derivative, the quadratic convergence of the Newton-Raphson method, which this unit only observed in examples.