The image contains mathematical tasks related to the analysis of functions and their properties. I’ll provide a structured approach to tackle these questions. Let’s address the key tasks systematically.

Task A

1. Study the variation of $f(x)$:

• $f(x) = -4x^3 - 4x - 1$
• Compute $f'(x)$ and identify the intervals where $f(x)$ is increasing or decreasing by solving $f'(x) = 0$.
• Analyze the sign of $f'(x)$ and provide a table of variations.

2. Tangent equation:

• Write the tangent equation for $f(x)$ at specific points $x = -0.5$ and $x = 1$.
• Use the formula for the tangent at $x = a$: $y = f'(a)(x - a) + f(a)$

Task B

1. Study of $g(x)$:

• $g(x) = x - \frac{1}{x}$
• Compute $g'(x)$, determine the critical points, and analyze the variation of $g(x)$.

2. Asymptotes:

• Analyze the behavior of $g(x)$ as $x \to \pm\infty$.
• Check for horizontal, vertical, or oblique asymptotes.

Approach

I can provide detailed steps for any of these tasks. Which one would you like me to start with, or would you prefer a specific problem fully solved?

Here are five relative questions to extend your understanding:

1. How do we calculate the critical points of a function?
2. What is the role of the derivative in studying the variation of functions?
3. How can we determine the equation of a tangent to a curve?
4. What are the mathematical steps to identify asymptotes of a rational function?
5. Why is the behavior of a function at infinity important in graphing?

Tip: Always graph the function to confirm the results of your algebraic analysis visually.