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Keywords:
Darboux function; mean value theorem; continuous function; integrable function; differentiable function; arithmetic mean; geometric mean; harmonic mean
Darboux function; mean value theorem; continuous function; integrable function; differentiable function; arithmetic mean; geometric mean; harmonic mean
Summary:
The aim of the paper is to present some mean value theorems obtained as consequences of the intermediate value property. First, we will prove that any nonextremum value of a Darboux function can be represented as an arithmetic, geometric or harmonic mean of some different values of this function. Then, we will present some extensions of the Cauchy or Lagrange Theorem in classical or integral form. Also, we include similar results involving divided differences. The paper was motivated by some problems published in mathematical journals.
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