Exercice 3

Exercice 3

  • Exercice 3 :

    On considère la fonction numérique g définie sur ]0, +∞[ par g(x) = 2√x – 2 – ln x

    a- Montrer que pour tout x de [1, +∞[, g'(x) = √x – 1/x
    b- Montrer que g est croissante sur [1, +∞[
    c- En déduire que pour tout x de [1, +∞[, 0 ≤ ln x ≤ 2√x (remarquer que 2√x – 2 ≤ 2√x)
    d- Montrer que pour tout x de [1, +∞[, 0 ≤ (ln x)3/x2 ≤ 8/√x et en déduire lim x →+∞ (ln x)3/x2

    2- a- Montrer que la fonction G : x ⟼ x(-1 + 4/3√x – ln x) est une primitive de g sur ]0, +∞[
    b- Calculer l’intégrale 4∫1 g(x)dx.

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