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Exercise Overview:

• Section A:

  1. Study the variations of $f(x) = 4x^4 - 4x^3 - 1$.
  2. Show the existence of $\alpha$ and $\beta$ with specified properties.
  3. Write the equation of the tangent to $(C)$ at a specified point.
  4. Analyze the restriction of $f$ on $]-\infty, -\frac{1}{4}]$ and demonstrate certain conditions.

• Section B:

  1. For $g(x) = x - \sqrt{x - 1}$, show the existence of an asymptote and position it.
  2. Study the function $t(x) = x^3 - x + 1$, establishing variations and intersections.

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