Mathematics RU

Practice · Chapter 27

Extrema

Critical points, maxima and minima from the sign of the derivative, and the largest or smallest value of a function on a closed interval.

How to solve it

At a maximum point a function stops increasing and starts decreasing; at a minimum point, the other way round. Increasing and decreasing show in the sign of the derivative, so extrema are found among the points where the derivative is zero or does not exist — the critical points.

Step by step

  1. Find the derivative and the domain of the function.
  2. Critical points: solve $f'(x) = 0$ and add the points where the derivative does not exist (while the function is defined).
  3. Find the sign of the derivative between them. Plus to minus is a maximum, minus to plus a minimum.
  4. The largest and smallest values on an interval: evaluate the function at the critical points inside the interval and at the ends, then pick.
The necessary condition: a horizontal tangent. The sufficient one: increasing turns into decreasing. A change $- \to +$ gives a minimum; no change, no extremum. Example: $f(x) = x + \frac{9}{x}$, $f'(x) = 1 - \frac{9}{x^2} = 0$ at $x = \pm 3$. Left of $-3$ the derivative is positive, right of it negative: $x = -3$ is a maximum point.

Common mistakes

  • Confusing the maximum point with the maximum value: the point is $x$, the value is $f(x)$. Check what is asked.
  • Forgetting the ends of the interval — the largest value is often there.
  • Using critical points outside the interval.
  • Not checking the sign change: $f(x) = x^3$ has a zero derivative at zero but no extremum.

Example

Mathematics

Chapters

Introduction

  1. 0 What mathematics is

I Numbers

  1. 1 Counting
  2. 2 Zero and minus
  3. 3 Primes
  4. 4 Divisibility
  5. 5 Fractions
  6. 6 Root two

II Algebra

  1. 7 Letters
  2. 8 Equations
  3. 9 Functions and graphs
  4. 10 Quadratic equations
  5. 11 Systems and inequalities
  6. 12 Powers and logarithms
  7. 13 Sequences
  8. 14 Polynomials
  9. 15 Complex numbers

III Geometry

  1. 16 Euclid's Elements
  2. 17 The triangle
  3. 18 Pythagoras' theorem
  4. 19 The circle and π
  5. 20 Area and volume
  6. 21 Trigonometry
  7. 22 Vectors
  8. 23 Symmetry
  9. 24 Conic sections

IV Calculus

  1. 25 Limits
  2. 26 The derivative
  3. 27 Extrema
  4. 28 The integral
  5. 29 The number e
  6. 30 Series
  7. 31 Differential equations
  8. 32 Several variables
  9. 33 Fourier series
  10. 34 Complex analysis

V Linear algebra

  1. 35 Matrices
  2. 36 Gaussian elimination
  3. 37 Vector spaces
  4. 38 Eigenvectors
  5. 39 Least squares and SVD

VI Structures

  1. 40 Groups
  2. 41 Remainders and ciphers
  3. 42 Rings and fields
  4. 43 Galois theory
  5. 44 Zeta and curves

VII Chance & data

  1. 45 Combinatorics
  2. 46 Graphs
  3. 47 Probability
  4. 48 Random variables
  5. 49 Statistics
  6. 50 Markov and information

VIII Foundations

  1. 51 Logic and sets
  2. 52 Infinities
  3. 53 Rigorous analysis
  4. 54 Measure and integral
  5. 55 Functional analysis
  6. 56 Gödel and Turing

IX Horizons

  1. 57 Topology
  2. 58 Non-Euclidean geometries
  3. 59 Chaos and fractals
  4. 60 The frontier