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### 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

Consider the function f(x) = 2 ln(x) / x, defined for x > 0. a) Study the maxima, minima, and regions of increase/decrease. b) Does this function have asymptotes? Make a sketch of its graph. c) Calculate the equation of the tangent line to the graph of y = f(x) at x = 1.

Solve a calculus problem involving maxima, minima, and the equation of the tangent line for the function f(x) = 2ln(x)/x. This detailed solution includes derivative rules, the first and second derivative tests, and a step-by-step method for finding asymptotes and critical points.

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