Derivatives and Functions’ Variation

  • Titu Andreescu
  • Cristinel Mortici
  • Marian Tetiva


The derivatives of a differentiable function \(f: [a,b] \rightarrow \mathbb{R}\) give us basic information about the variation of the function. For instance, it is well-known that if f  ≥ 0, then the function f is increasing and if f  ≤ 0, then f is decreasing. Also, a function defined on an interval having the derivative equal to zero is in fact constant. All of these are consequences of some very useful theorems due to Fermat, Cauchy, and Lagrange. Fermat’s theorem states that the derivative of a function vanishes at each interior extremum point of f. The proof is not difficult: suppose that x0 is a local extremum, let us say a local minimum. Then f(x0 + h) − f(x0) ≥ 0 for all h in an open interval (−δ, δ). By dividing by h and passing to the limit when h approaches 0, we deduce that f (x0) ≥ 0 (for h > 0) and f (x0) ≤ 0 (for h < 0); thus f (x0) = 0.


Real Number Differentiable Function Positive Real Number Local Extremum Riemann Zeta Function 
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© Springer Science+Business Media LLC 2017

Authors and Affiliations

  • Titu Andreescu
    • 1
  • Cristinel Mortici
    • 2
  • Marian Tetiva
    • 3
  1. 1.University of Texas at Dallas Natural Sciences and MathematicsRichardsonUSA
  2. 2.Valahia University of TargovisteTargovisteRomania
  3. 3.Gheorghe Rosca Codreanu National CollegeBarladRomania

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