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how do i feel up the « tableau de variation de la fonction »

To fill in the **"tableau de variation de la fonction"** (variation table of the function) for the given function \( f(x) = -x^2 + 6x + 5 \), we need to analyze the behavior of the function.

### Steps to complete the table:

1. **Find the critical points**:
   The function is a quadratic in the form \( f(x) = -x^2 + 6x + 5 \), which is a downward-opening parabola because the coefficient of \( x^2 \) is negative. The maximum occurs at the vertex. The x-coordinate of the vertex is given by:
   \[
   x_v = \frac{-b}{2a} = \frac{-6}{2(-1)} = 3
   \]
   So, the function reaches its maximum at \( x = 3 \).

2. **Determine the intervals of increasing and decreasing**:
   - For \( x < 3 \), the function is **increasing** because the parabola slopes upward.
   - For \( x > 3 \), the function is **decreasing** because the parabola slopes downward.

3. **Calculate the function values** at critical points:
   - At \( x = 3 \), the function reaches its maximum. Plugging in \( x = 3 \) into \( f(x) \):
   \[
   f(3) = -(3)^2 + 6(3) + 5 = -9 + 18 + 5 = 14
   \]
   So, the maximum value of the function is 14.

4. **Complete the table**:
   - The function increases on the interval \( [-1, 3] \).
   - The function decreases on the interval \( [3, 7] \).
   - The maximum value occurs at \( x = 3 \) and is 14.

The completed variation table would look like this:

| \( x \)   | \(-1\)   |            | \( 3 \)           |            | 7      |
|-----------|----------|------------|-------------------|------------|--------|
| \( f(x) \) | \( -2 \) | \(\nearrow\) (increases) | \( 14 \) (max) | \(\searrow\) (decreases) | \( -2 \) |

Here’s what this means:
- \( f(x) \) increases from \( x = -1 \) to \( x = 3 \), reaching a maximum value of 14 at \( x = 3 \).
- \( f(x) \) decreases from \( x = 3 \) to \( x = 7 \).

Let me know if you want further clarifications on this, or if you'd like help with anything else!

### Relative Questions:
1. How do you determine the vertex of a quadratic function?
2. Why does a negative \( x^2 \) term indicate a downward-opening parabola?
3. What is the significance of the increasing and decreasing intervals of a function?
4. Can a quadratic function have more than one maximum or minimum point?
5. How do we calculate the value of a function at specific points?

### Tip:
Always check the sign of the quadratic term (\( x^2 \)) to quickly determine whether the function opens upward or downward, which helps in finding whether you have a maximum or minimum at the vertex.

How do I fill in the tableau de variation de la fonction for the quadratic function f(x) = -x^2 + 6x + 5?

Learn how to complete the tableau de variation for the quadratic function f(x) = -x^2 + 6x + 5. This problem covers finding the vertex, analyzing increasing and decreasing intervals, and graphing the function on a specific domain.

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