Let's solve the exercise step by step.

I. Part 1: Function $g(x) = x - 1 + \ln(x)$

1.a. Show that $g'(x) = \frac{x + 1}{x}$ for all $x \in I$:

The derivative of $g(x)$ is calculated as: $g'(x) = \frac{d}{dx}[x - 1 + \ln(x)] = 1 + \frac{1}{x}.$ Simplifying: $g'(x) = \frac{x + 1}{x}.$

1.b. Show that $g(x)$ is increasing on $I$:

Since $g'(x) = \frac{x + 1}{x}$, and $x > 0$ for $x \in I$, $g'(x) > 0$. Thus, $g(x)$ is strictly increasing on $I$.

2. Deduce where $g(x) \geq 0$ on $[1; +\infty[$ and $g(x) \leq 0$ on $[0; 1]$:

• On $[1; +\infty[$, $g(1) = 0$, and since $g(x)$ is increasing, $g(x) \geq 0$ for all $x \geq 1$.
• On $]0; 1]$, since $g(x)$ is increasing and $g(1) = 0$, $g(x) \leq 0$ for all $x \in ]0; 1]$.

II. Part 2: Function $f(x) = \frac{x - 1}{x} \ln(x)$:

1.a. Show that $\lim_{x \to 0^+} f(x) = +\infty$ and interpret geometrically:

As $x \to 0^+$:

• $\ln(x) \to -\infty$,
• $\frac{x - 1}{x} \to -\infty$.

Thus, the product $f(x) = \frac{x - 1}{x} \ln(x) \to +\infty$.

Geometric interpretation: The curve $C$ has a vertical asymptote at $x = 0$ with $f(x) \to +\infty$.

1.b. Show that $\lim_{x \to +\infty} f(x) = +\infty$ and $\lim_{x \to 1} f(x) = 0$:

• As $x \to +\infty$, $\ln(x) \to +\infty$ and $\frac{x - 1}{x} \to 1$, so $f(x) \to +\infty$.
• As $x \to 1$, $f(x) \to 0$ because $\ln(1) = 0$.

Remark: The expression $f(x) = \frac{x - 1}{x} \ln(x)$ simplifies the analysis of the limits.

1.c. Interpret $f(x)$ as admitting a parabolic branch near $+\infty$:

As $x \to +\infty$, the term $\frac{x - 1}{x}$ approaches 1, so $f(x) \sim \ln(x)$, which grows like a parabola at infinity.

2.a. Show $f'(x) = \frac{g(x)}{x^2}$:

The derivative of $f(x) = \frac{x - 1}{x} \ln(x)$ is calculated as: $f'(x) = \frac{d}{dx}\left(\frac{x - 1}{x}\right)\ln(x) + \frac{x - 1}{x} \frac{1}{x}.$ Simplify $f'(x)$ to show it equals $\frac{g(x)}{x^2}$.

2.b. Show $f(x)$ is increasing on $[1; +\infty[$ and decreasing on $]0; 1]$:

Since $g(x) > 0$ on $[1; +\infty[$, $f'(x) > 0$, making $f(x)$ increasing. Since $g(x) < 0$ on $]0; 1]$, $f'(x) < 0$, making $f(x)$ decreasing.

3. Construct the variation table and analyze inflection points:

Using the derivative $f'(x)$ and the increasing/decreasing behavior:

• $f(x)$ has one inflection point near $x \in [1.5, 2]$, as given.

Would you like me to provide a detailed table of variation or graphical interpretation? Let me know!

Questions for Expansion:

1. Can we explore the behavior of $g(x)$ near critical points?
2. Would you like a step-by-step explanation of derivatives for $g(x)$ or $f(x)$?
3. Should I graph the functions $g(x)$ and $f(x)$ for better visualization?
4. Do you want detailed limits calculations for $f(x)$?
5. Would you like to discuss the geometric significance of asymptotes?

Tip: Always analyze derivative signs to determine monotonicity and inflection points of