Math Problem Statement

On considère la fonction numérique h de la variable réelle x définie sur ]0; +∞[ par : h(x) = (ln(x))^3 - 3ln(x) et sa courbe représentative dans un repère orthonormé. L'exercice comporte plusieurs parties comme la vérification de certaines limites, le calcul des dérivées, et l'étude du tableau de variations de la fonction.

Solution

It looks like you've uploaded an image of a math exercise involving a function h(x)=(ln(x))33ln(x)h(x) = (\ln(x))^3 - 3\ln(x). The exercise involves multiple parts, including limits, derivatives, solving an equation, and analyzing the function’s variations.

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Math Problem Analysis

Mathematical Concepts

Limits
Derivatives
Function Variation
Logarithmic Functions

Formulas

h(x) = (ln(x))^3 - 3ln(x)
lim(x→∞) ln(x)^3 / x
h'(x) = 3x * (ln(x)-1) * (ln(x)+1)

Theorems

Limit Theorem
Derivative Rules
Monotonicity Test

Suitable Grade Level

Grades 11-12