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Let's break down **Exercise 8** from the image.

### Part A:
Consider the polynomial function \( g(x) = -4x^3 + 4x \).

1. **Study the variations of \( g \) over \( \mathbb{R} \) and draw its table of variations.**  
   We need to find the derivative \( g'(x) \) to determine where the function increases or decreases.
   
2. **Determine the abscissa of the point of intersection between (C) and (O).**  
   Here, "abscissa" refers to the x-coordinate of the points where the curve intersects the x-axis. We solve \( g(x) = 0 \) to find these values.
   
3. **Draw the curve (C).**  
   Sketch the curve of \( g(x) \), noting important features like critical points and points of intersection.

4. **Deduce graphically the sign of \( g(x) \).**  
   Use the graph to identify where \( g(x) \) is positive or negative.

### Part B:
Let \( f(x) = -x^4 + 2x^2 - 1 \).

1. **Calculate the limits of \( f \) at the boundaries of \( \mathbb{R} \).**  
   We need to compute \( \lim_{x \to \infty} f(x) \) and \( \lim_{x \to -\infty} f(x) \).

2. **Verify that \( f'(x) = g(x) \).**  
   We calculate \( f'(x) \) and check if it equals \( g(x) \).

3. **Set up the table of variations of \( f \).**  
   Using the critical points from \( f'(x) = 0 \), determine where \( f(x) \) increases or decreases.

4. **Trace (C') the representative curve of \( f \) in a new system.**  
   Plot the curve of \( f(x) \) in a coordinate system.

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Would you like me to solve any specific parts of this exercise? 

### Relative Questions:
1. What is the process for determining the critical points of a function?
2. How do we compute limits at infinity for polynomial functions?
3. What is the significance of the table of variations in understanding a function's behavior?
4. How do we interpret the sign of a function from its graph?
5. How can we verify if one function is the derivative of another?

### Tip:
When studying the variation of a function, always start by finding the critical points and determining whether the function is increasing or decreasing around these points.

Consider the polynomial function g defined over R by g(x) = -4x^3 + 4x. Study its variations, find the point of intersection with the x-axis, and draw the representative curve. Then analyze the function f(x) = -x^4 + 2x^2 - 1 and verify if its derivative matches g(x).

This exercise explores the behavior of polynomial functions, focusing on g(x) = -4x^3 + 4x. It includes studying variations, identifying intersection points, and drawing the representative curve. The function f(x) = -x^4 + 2x^2 - 1 is analyzed to verify that its derivative matches g(x).

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