Let's analyze and solve the questions from the image step by step:

1. Function Overview

We are given the function:

$f(x) = 2 \frac{\ln(x)}{x}, \quad \text{with} \quad x > 0.$

a) Study the maxima, minima, and regions of increase/decrease

1.1 Critical Points:

To find the critical points (maxima or minima), we need to calculate the derivative of $f(x)$:

$f'(x) = \frac{d}{dx} \left( 2 \frac{\ln(x)}{x} \right).$

Using the quotient rule for derivatives $\left( \frac{u(x)}{v(x)} \right)' = \frac{u'(x)v(x) - u(x)v'(x)}{v(x)^2}$, where $u(x) = 2 \ln(x)$ and $v(x) = x$, we get:

$f'(x) = \frac{2 \left( \frac{1}{x} \cdot x - \ln(x) \cdot 1 \right)}{x^2} = \frac{2 (1 - \ln(x))}{x^2}.$

Set $f'(x) = 0$ to find critical points:

$1 - \ln(x) = 0 \quad \Rightarrow \quad \ln(x) = 1 \quad \Rightarrow \quad x = e.$

So, $x = e$ is a critical point. To determine whether it is a maximum or minimum, we check the second derivative $f''(x)$.

1.2 Second Derivative:

Differentiate $f'(x) = \frac{2 (1 - \ln(x))}{x^2}$:

[ f''(x) = \frac{d