The image contains a set of mathematical problems related to the study of the relative position of a curve and its tangents. Here’s a breakdown of the problems:

1. Problem 1: Show that the reduced equation of the tangent (T) to the curve $\mathcal{C}$ at $x = 2$ is $y = 3x + 3$.

2. Problem 2: Show that studying the relative position of the curve $\mathcal{C}$ and its tangent (T) at $x = 2$ amounts to studying the sign of the function: $\frac{x^3 - 3x^2 + 4}{x - 1}.$

3. Problem 3: Show that for any real $x$, $x^3 - 3x^2 + 4 = (x - 2)^2 (x + 1).$

4. Problem 4: Deduce the relative position of the curve $\mathcal{C}$ and its tangent (T) at $x = 2$ on the interval $[1.5; 5]$.

Solution Outline

To approach these problems, we can follow these steps:

1. Finding the Tangent Equation:

   • Given $x = 2$, find the derivative $f'(x)$ to determine the slope of the tangent.
   • Use the point-slope form of the tangent line at $x = 2$ to derive its equation.

2. Studying the Sign of the Function:

   • Evaluate the given function $\frac{x^3 - 3x^2 + 4}{x - 1}$ to understand how it represents the difference between the curve and the tangent line.

3. Factorization:

   • Factor $x^3 - 3x^2 + 4$ to show that it equals $(x - 2)^2 (x + 1)$.

4. Analyzing the Relative Position:

   • Determine the sign of the function over the interval $[1.5, 5]$ to understand where the curve lies relative to the tangent.

Would you like a detailed explanation of how to solve each of these steps?

Further Questions

1. How do we find the derivative of a function to determine the slope of a tangent line?
2. Why is it necessary to factor $x^3 - 3x^2 + 4$ when analyzing the curve?
3. What does the sign of $\frac{x^3 - 3x^2 + 4}{x - 1}$ tell us about the curve and tangent?
4. How can we interpret the intervals where the curve is above or below the tangent line?
5. What is the significance of the interval $[1.5; 5]$ in this problem?

Tip

When analyzing the position of a curve relative to its tangent, consider the points of intersection and the concavity of the curve to understand their geometric relationship better.