https://r2.erweima.ai/mathbot/image/172625418783803mluau0.jpg

Let's go through the exercise step by step.

### Given:
The function \( f(x) \) is defined by:
\[
f(x) = -2(x - 1)^2 + 8
\]

---

### 1. **Développer \( f(x) \):**
We expand the expression for \( f(x) \).

\[
f(x) = -2(x - 1)^2 + 8
\]
Expand \( (x - 1)^2 \):
\[
(x - 1)^2 = x^2 - 2x + 1
\]
Thus:
\[
f(x) = -2(x^2 - 2x + 1) + 8 = -2x^2 + 4x - 2 + 8 = -2x^2 + 4x + 6
\]

So, the expanded form of \( f(x) \) is:
\[
f(x) = -2x^2 + 4x + 6
\]

---

### 2. **Justifier que pour tout réel \( x \), on a \( f(x) = (2x + 2)(-x + 3) \):**
We need to verify that \( f(x) = -2x^2 + 4x + 6 \) can be factored as \( (2x + 2)(-x + 3) \).

First, expand \( (2x + 2)(-x + 3) \):
\[
(2x + 2)(-x + 3) = 2x(-x + 3) + 2(-x + 3) = -2x^2 + 6x - 2x + 6 = -2x^2 + 4x + 6
\]

This matches the expanded form of \( f(x) \). Therefore, the factorization is correct:
\[
f(x) = (2x + 2)(-x + 3)
\]

---

### 3a. **Dresser le tableau de signes de \( f(x) \):**
We analyze the signs of \( f(x) = (2x + 2)(-x + 3) \) by finding the roots.

- \( 2x + 2 = 0 \) gives \( x = -1 \)
- \( -x + 3 = 0 \) gives \( x = 3 \)

Thus, the function changes sign at \( x = -1 \) and \( x = 3 \).

Now we check the signs in each interval:
- For \( x < -1 \), both factors are negative, so \( f(x) > 0 \).
- For \( -1 < x < 3 \), \( 2x + 2 > 0 \) and \( -x + 3 > 0 \), so \( f(x) < 0 \).
- For \( x > 3 \), both factors are positive, so \( f(x) > 0 \).

The sign table is:
\[
\begin{array}{|c|c|c|c|}
\hline
x & (-\infty, -1) & (-1, 3) & (3, +\infty) \\
\hline
f(x) & + & - & + \\
\hline
\end{array}
\]

---

### 3b. **En déduire les solutions de l'inéquation \( f(x) \geq 0 \):**
From the sign table, we can see that \( f(x) \geq 0 \) when \( x \in (-\infty, -1] \cup [3, +\infty) \).

---

### 4. **Déterminer l'extremum (maximum ou minimum) de la fonction \( f \):**
Since \( f(x) = -2(x - 1)^2 + 8 \) is a quadratic function with a negative leading coefficient, it has a maximum at the vertex. The vertex is given by \( x = 1 \).

Substitute \( x = 1 \) into \( f(x) \):
\[
f(1) = -2(1 - 1)^2 + 8 = 8
\]
Thus, the maximum value of \( f(x) \) is 8 at \( x = 1 \).

---

### 5. **Dresser le tableau des variations de la fonction \( f \):**
The function is a parabola that opens downwards. From the previous analysis:
- \( f(x) \) increases on \( (-\infty, 1] \) and decreases on \( [1, +\infty) \).

The variation table is:
\[
\begin{array}{|c|c|c|c|}
\hline
x & (-\infty, 1) & 1 & (1, +\infty) \\
\hline
f(x) & \text{increasing} & 8 & \text{decreasing} \\
\hline
\end{array}
\]

---

### 6. **Tracer l'allure de la courbe représentative:**
You are asked to sketch the curve of \( f(x) \), using the previously found information (signs, extremum, and variations).

The key points to plot are:
- \( f(-1) = 0 \), \( f(3) = 0 \) (roots),
- Maximum at \( x = 1 \), \( f(1) = 8 \),
- Asymptotic behavior as \( x \to \pm \infty \).

---

Would you like a more detailed explanation or help with specific parts of the exercise?

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### Related questions:
1. What is the general form of a quadratic function, and how can its extremum be found?
2. How do you determine the roots of a quadratic function using factoring?
3. What is the relationship between the sign of the leading coefficient and the direction of the parabola?
4. How do you create a variation table from the derivative of a function?
5. How would you solve a quadratic inequality graphically?

#### Tip:
For quadratic functions in vertex form \( f(x) = a(x - h)^2 + k \), the vertex \( (h, k) \) gives the extremum, and the sign of \( a \) determines whether it is a maximum or minimum.

Une fonction f est définie sur l'intervalle ]-∞;+∞[ par : f(x) = -2(x - 1)^2 + 8. 1. Développer f(x). 2. Justifier que pour tout réel x, on a : f(x) = (2x + 2)(-x + 3). 3a. Dresser le tableau de signes de f(x). 3b. En déduire les solutions de l'inéquation f(x) ≥ 0. 4. Déterminer l'extremum (maximum ou minimum) de la fonction f. 5. Dresser le tableau des variations de la fonction f. 6. Tracer l'allure de la courbe représentative de la fonction f.

In this exercise, we explore the quadratic function f(x) = -2(x - 1)^2 + 8 by expanding it, analyzing its sign, finding its maximum value, and plotting its graph. The exercise covers key topics like quadratic factorization, sign tables, and extremum determination, suitable for high school students.

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