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The problem provided seems to be focused on studying the relative position of curves, specifically the function \( f(x) \) and its tangent \( T \). Let me transcribe and summarize the key tasks from the image for clarity:

**Part C: Étude de la position relative de la courbe de la fonction \( f \) et de l'une de ses tangentes**

1. **Montrer que l'équation réduite de la tangente \( T \) à la courbe \( \mathcal{C} \) en \( x = 2 \) est :**
   \[
   y = 3x + 3
   \]

2. **Montrer qu'étudier la position relative de \( \mathcal{C} \) et \( T \) revient à étudier le signe de :**
   \[
   \frac{x^3 - 3x^2 + 4}{(x - 2)^2 (x + 1)}
   \]

3. **Montrer que, pour tout réel \( x \), \( x^3 - 3x^2 + 4 = (x - 2)^2 (x + 1) \).**

4. **En déduire alors la position relative de la courbe \( \mathcal{C} \) et de sa tangente \( T \) en \( x = 2 \) sur l'intervalle \( [1.5, 5] \).**

The problem is asking to study the position of the curve \( \mathcal{C} \) defined by the function \( f(x) = x^3 - 3x^2 + 4 \) and its tangent at \( x = 2 \), with the following steps:

- Show that the tangent equation at \( x = 2 \) is \( y = 3x + 3 \).
- Study the sign of the difference between \( f(x) \) and the tangent to understand where the curve is above or below the tangent.
- Prove that the cubic function can be factored as \( (x - 2)^2 (x + 1) \).
- Analyze the relative position of the curve and tangent in the interval \( [1.5, 5] \).

Would you like me to guide you through these steps in detail or solve one of the specific points?

Étude de la position relative de la courbe de la fonction f et de l'une de ses tangentes. Montrer que l'équation réduite de la tangente T à la courbe C en x = 2 est y = 3x + 3. Étudier la position relative de C et T en étudiant le signe de la différence de leurs fonctions.

This problem explores the relative position of a cubic curve and its tangent at a specific point. We derive the equation of the tangent, analyze the polynomial's behavior, and factor it to study the relative position in a given interval.

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