Derivatives and Functions’ Variation

Abstract

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 x ₀ is a local extremum, let us say a local minimum. Then f(x ₀ + h) − f(x ₀) ≥ 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 ^′(x ₀) ≥ 0 (for h > 0) and f ^′(x ₀) ≤ 0 (for h < 0); thus f ^′(x ₀) = 0.