https://r2.erweima.ai/mathbot/image/1727182603130ukma9ofb.jpg

The problem you're working on defines the function \( f(x) \) as:

\[
f(x) = (2m + \frac{7}{4})x^2 + (3m + 2)x + m
\]

and asks you to do two things:

1. **Determine all values of the parameter \( m \)** for which the function \( f \) is a second-degree polynomial that has a minimum and crosses the x-axis at exactly two distinct points.

2. **Identify the images of the function \( f \) that do not depend on the parameter \( m \)**, and for each image, give the corresponding antecedent (pre-image).

---

Let's start with **Question 1**:

We have the quadratic function \( f(x) = ax^2 + bx + c \), where:
- \( a = 2m + \frac{7}{4} \)
- \( b = 3m + 2 \)
- \( c = m \)

### Conditions for the function:

1. **Minimum condition**: 
   For the quadratic to have a minimum, the coefficient \( a \) of \( x^2 \) must be positive. Thus:

   \[
   2m + \frac{7}{4} > 0
   \]

   Solving for \( m \):

   \[
   2m > -\frac{7}{4} \quad \Rightarrow \quad m > -\frac{7}{8}
   \]

2. **Two distinct roots**:
   A quadratic function has two distinct roots if the discriminant \( \Delta \) is positive. The discriminant is given by:

   \[
   \Delta = b^2 - 4ac
   \]

   Substituting the values of \( a \), \( b \), and \( c \):

   \[
   \Delta = (3m + 2)^2 - 4(2m + \frac{7}{4})m
   \]

   Let's simplify this discriminant and solve for \( m \).

I'll calculate this part for you.The discriminant simplifies to:

\[
\Delta = m^2 + 5m + 4
\]

For the quadratic to have two distinct roots, we require \( \Delta > 0 \). So, we need to solve the inequality:

\[
m^2 + 5m + 4 > 0
\]

Let's solve this inequality to determine the values of \( m \).The roots of the discriminant equation \( m^2 + 5m + 4 = 0 \) are \( m = -4 \) and \( m = -1 \). 

Thus, to satisfy \( \Delta > 0 \), the parameter \( m \) must lie in the intervals:

\[
m \in (-\infty, -4) \cup (-1, \infty)
\]

### Combining the conditions:
- From the minimum condition: \( m > -\frac{7}{8} \)
- From the discriminant: \( m \in (-\infty, -4) \cup (-1, \infty) \)

The solution for \( m \), satisfying both conditions, is:

\[
m \in (-1, \infty)
\]

### For **Question 1**: The parameter \( m \) must be greater than \( -1 \) to ensure a minimum and two distinct roots.

---

Now, let's move on to **Question 2** about the images of the function that do not depend on \( m \). I'll analyze that next.### **Question 2: Finding the images of \( f(x) \) that do not depend on \( m \)**

To answer this, we need to check under which conditions the value of \( f(x) \) is independent of \( m \). This happens if the terms involving \( m \) in the expression for \( f(x) \) cancel out.

Given:
\[
f(x) = (2m + \frac{7}{4})x^2 + (3m + 2)x + m
\]

Let's find specific values of \( x \) such that \( f(x) \) is independent of \( m \). For this to happen, the coefficients of \( m \) in the expression must all vanish.

I'll solve for \( x \) values where the terms containing \( m \) disappear.It turns out that there are no specific values of \( x \) that make the function independent of \( m \) in general. This means that \( f(x) \) always depends on \( m \) for any value of \( x \).

Thus, **Question 2** has no solutions for \( x \) where \( f(x) \) is independent of \( m \).

---

To summarize:

1. For the function to have a minimum and two distinct roots, the parameter \( m \) must satisfy \( m > -1 \).
2. There are no specific values of \( x \) where the function \( f(x) \) is independent of the parameter \( m \).

Would you like more details on any part of this solution?

---

Here are 5 related questions for further exploration:
1. How do we find the vertex of a quadratic function like \( f(x) \)?
2. What are the implications if \( m = -1 \) or \( m = -4 \)?
3. How can you graphically verify the two distinct roots of the quadratic function?
4. How would you solve this if the discriminant \( \Delta = 0 \) instead of being positive?
5. How does the quadratic formula help in determining the exact roots of the function?

**Tip**: When analyzing quadratic functions, always remember to check the sign of the leading coefficient \( a \) to determine if the parabola opens upwards (minimum) or downwards (maximum).

Soit m un nombre réel. Pour x ∈ ℝ, on définit la fonction f par : f(x) = (2m + 7/4)x^2 + (3m + 2)x + m. Déterminer toutes les valeurs du paramètre m pour que la fonction f soit un polynôme du second degré admettant un minimum et dont la courbe coupe l’axe des abscisses en exactement 2 endroits distincts. Quelles sont les images de la fonction f ne dépendant pas du paramètre m ? Pour chacune, en donner l’antécédent.

Solve the quadratic function f(x) = (2m + 7/4)x^2 + (3m + 2)x + m for values of m where it has a minimum and exactly two distinct roots. Understand the discriminant and conditions for minimum.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation