Math 2.0

Real functions Proof

Proof of the first derivative test

Builds on A function with nonnegative derivative is increasingDifferentiable functions are continuousLocal minimumLocal maximumConstant multiple rule

Proof

We apply the monotonicity test on each side of cc.

  1. Assume f′(x)≤0f'(x) \le 0 for c−δ<x<cc - \delta < x < c and f′(x)≥0f'(x) \ge 0 for c<x<c+δc < x < c + \delta.
  2. The function ff is continuous at each x≠cx \ne c, since it is differentiable there. It is continuous at cc by hypothesis.
  3. Let x∈(c,c+δ)x \in (c, c + \delta). Then ff is continuous on [c,x][c, x] and differentiable on (c,x)(c, x) with f′≥0f' \ge 0 there.
  4. By the nonnegative derivative test on the interval [c,x][c, x], ff is increasing there. So f(c)≤f(x)f(c) \le f(x).
  5. Let x∈(c−δ,c)x \in (c - \delta, c). The function −f-f is continuous on [x,c][x, c] and differentiable on (x,c)(x, c).
  6. By the constant multiple rule, (−f)′=−f′≥0(-f)' = -f' \ge 0 on (x,c)(x, c).
  7. By the nonnegative derivative test, −f-f is increasing on [x,c][x, c]. So −f(x)≤−f(c)-f(x) \le -f(c), that is, f(c)≤f(x)f(c) \le f(x).
  8. Steps 4 and 7 give f(x)≥f(c)f(x) \ge f(c) for all xx with ∣x−c∣<δ|x - c| < \delta. So ff has a local minimum at cc.
  9. Now assume the reversed inequalities: f′≥0f' \ge 0 on (c−δ,c)(c - \delta, c) and f′≤0f' \le 0 on (c,c+δ)(c, c + \delta).
  10. Then −f-f satisfies the hypotheses of step 1, since (−f)′=−f′(-f)' = -f'. By step 8, −f-f has a local minimum at cc.
  11. So −f(x)≥−f(c)-f(x) \ge -f(c) near cc, that is, f(x)≤f(c)f(x) \le f(c). Thus ff has a local maximum at cc.

Idea

A function that does not increase up to cc and does not decrease after cc is lowest at cc. The mean value theorem, through the monotonicity test, turns the sign conditions on f′f' into these monotonicity facts. Continuity at cc lets the test include the point cc itself.

Reading path

To understand this page from the ground up, read these 140 pages in order. The foundations come first. The path takes the proofs Proof of the limit of a sum of sequences, Proof of Rolle's theorem, Proof of the approximation property of the supremum, Proof that every sequence has a monotone subsequence, by peaks, Proof of the commutativity of addition by induction, Proof of the epsilon-delta form of continuity, Proof that there is a natural number n with 1/n below any positive epsilon, Proof that a function with nonnegative derivative is increasing, Proof that the reals form an ordered field, with the Cauchy sequence construction, Proof of the double negation law by a truth table, Proof of the extreme value theorem by the Bolzano–Weierstrass theorem, Proof of the uniqueness of limits, Proof of strong induction from ordinary induction, Proof of the limit of a product of sequences, Proof that the absolute value is multiplicative, Proof of the squeeze theorem for sequences, Proof of the linearity of differentiation, Proof of the sequential characterization of limits, Proof of the triangle inequality for real numbers, Proof of the epsilon-N definition of the limit of a sequence, Proof of Fermat's theorem on stationary points, Proof that limits preserve non-strict inequalities, Proof that differentiable functions are continuous, Proof of the epsilon-delta form of the limit of a function, Proof of the well-ordering principle from strong induction, Proof of the Bolzano–Weierstrass theorem by a monotone subsequence, Proof of the Archimedean property in an Archimedean ordered field, Proof that a decreasing sequence bounded below converges, Proof that the reals are Archimedean from the completeness axiom, Proof that convergent sequences are bounded, Proof of the monotone convergence theorem for sequences, Proof that continuity is equivalent to sequential continuity, Proof of the boundedness theorem by Bolzano–Weierstrass, Proof of the mean value theorem from Rolle's theorem, Proof of the constant multiple rule.

  1. Construction of the reals from Cauchy sequences concept
  2. Definition by recursion concept
  3. Element definition
  4. Integers object
  5. Limit point of a set definition
  6. Local maximum definition
  7. Mathematical induction concept
  8. Natural numbers object
  9. Ordered field definition
  10. Peano axioms concept
  11. Proposition concept
  12. Rational numbers object
  13. Real numbers object
  14. Sequence definition
  15. Set bounded above definition
  16. Set definition
  17. Ring of integers statement
  18. Limit of a function definition
  19. Local minimum definition
  20. Archimedean ordered field definition
  21. Successor function definition
  22. Conjunction definition
  23. Disjunction definition
  24. Implication definition
  25. Negation definition
  26. Predicate definition
  27. The reals form a field statement
  28. Real sequence definition
  29. Subsequence definition
  30. Infimum definition
  31. Supremum definition
  32. Ordered pair definition
  33. Continuity at a point definition
  34. Derivative definition
  35. Addition of natural numbers definition
  36. Proof of the Archimedean property in an Archimedean ordered field proof
  37. The rationals form a field statement
  38. Biconditional definition
  39. The successor function is injective statement
  40. Zero is not a successor statement
  41. Truth table concept
  42. Modus ponens concept
  43. Universal quantifier definition
  44. Bounded sequence definition
  45. Limit of a sequence definition
  46. Monotone sequence definition
  47. Completeness axiom statement
  48. Cartesian product definition
  49. Continuous function definition
  50. Differentiable function definition
  51. Archimedean property statement
  52. Proof of the commutativity of addition by induction proof
  53. Order on the natural numbers definition
  54. Subset definition
  55. The rationals form an ordered field statement
  56. Propositional logic concept
  57. Mathematical proof concept
  58. Convergent sequence definition
  59. Sequential continuity definition
  60. Binary relation definition
  61. Order on the reals definition
  62. Commutativity of addition statement
  63. Proof of strong induction from ordinary induction proof
  64. Tautology definition
  65. Antisymmetric relation definition
  66. Reflexive relation definition
  67. Transitive relation definition
  68. Interval definition
  69. Strong induction statement
  70. Logical equivalence definition
  71. Preorder definition
  72. Proof of the epsilon-delta form of continuity proof
  73. Proof of the epsilon-delta form of the limit of a function proof
  74. Proof of the well-ordering principle from strong induction proof
  75. Proof of the double negation law by a truth table proof
  76. Partial order definition
  77. Epsilon-delta definition of continuity statement
  78. Epsilon-delta definition of a limit statement
  79. Well-ordering principle statement
  80. Double negation law statement
  81. Total order definition
  82. Proof by contradiction concept
  83. Proof that every sequence has a monotone subsequence, by peaks proof
  84. Proof that the reals form an ordered field, with the Cauchy sequence construction proof
  85. Every sequence has a monotone subsequence statement
  86. The reals form an ordered field statement
  87. Absolute value definition
  88. Proof of the approximation property of the supremum proof
  89. Proof of the triangle inequality for real numbers proof
  90. Proof that the absolute value is multiplicative proof
  91. Proof of the epsilon-N definition of the limit of a sequence proof
  92. Approximation property of the supremum statement
  93. Triangle inequality statement
  94. Absolute value is multiplicative statement
  95. Epsilon-N definition of the limit of a sequence statement
  96. Proof that the reals are Archimedean from the completeness axiom proof
  97. Proof of the limit of a sum of sequences proof
  98. Proof of the squeeze theorem for sequences proof
  99. Proof of the uniqueness of limits proof
  100. Proof that limits preserve non-strict inequalities proof
  101. Proof that differentiable functions are continuous proof
  102. The reals are Archimedean statement
  103. Proof that convergent sequences are bounded proof
  104. Proof of the monotone convergence theorem for sequences proof
  105. Differentiable functions are continuous statement
  106. Proof that there is a natural number n with 1/n below any positive epsilon proof
  107. Limit of a sum of sequences statement
  108. Uniqueness of limits statement
  109. Limits preserve non-strict inequalities statement
  110. Squeeze theorem statement
  111. Convergent sequences are bounded statement
  112. Monotone convergence theorem for sequences statement
  113. Proof of the limit of a product of sequences proof
  114. There is a natural number n with 1/n below any positive epsilon statement
  115. Proof that a decreasing sequence bounded below converges proof
  116. Proof of the sequential characterization of limits proof
  117. Decreasing sequence bounded below converges statement
  118. Proof that continuity is equivalent to sequential continuity proof
  119. Limit of a product of sequences statement
  120. Continuity is equivalent to sequential continuity statement
  121. Sequential characterization of limits statement
  122. Proof of the Bolzano–Weierstrass theorem by a monotone subsequence proof
  123. Proof of the linearity of differentiation proof
  124. Bolzano–Weierstrass theorem statement
  125. Linearity of differentiation statement
  126. Proof of the boundedness theorem by Bolzano–Weierstrass proof
  127. Proof of the constant multiple rule proof
  128. Boundedness theorem statement
  129. Constant multiple rule statement
  130. Proof of the extreme value theorem by the Bolzano–Weierstrass theorem proof
  131. Extreme value theorem statement
  132. Proof of Fermat's theorem on stationary points proof
  133. Fermat's theorem on stationary points statement
  134. Proof of Rolle's theorem proof
  135. Rolle's theorem statement
  136. Proof of the mean value theorem from Rolle's theorem proof
  137. Mean value theorem statement
  138. Proof that a function with nonnegative derivative is increasing proof
  139. A function with nonnegative derivative is increasing statement
  140. Proof of the first derivative test proof

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Derivative_test
About this page
TagWAXMH9PJX8
URIm2://WAXMH9PJX8
Kindproof
WritingW0 stub
MSC26A24
ProvesFirst derivative test
UsesA function with nonnegative derivative is increasing, Differentiable functions are continuous, Local minimum, Local maximum, Constant multiple rule
Changed4 November 2026

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