https://r2.erweima.ai/mathbot/image/17271816721504kmjl9ai.jpg

Let's analyze the problem.

The given function is:

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

### Question 1:
"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."

This asks for the values of \( m \) such that:

1. The function is a quadratic with a minimum.
2. The graph of the function intersects the x-axis at exactly two distinct points.

For the first condition, the function \( f(x) \) must open upwards, which implies the coefficient of \( x^2 \) (i.e., \( 2m + \frac{7}{4} \)) must be positive. This gives us:

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

For the second condition, the discriminant of the quadratic equation \( f(x) = 0 \) must be strictly positive to ensure two distinct roots. The discriminant \( \Delta \) for a quadratic equation \( ax^2 + bx + c \) is given by:

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

For our function \( f(x) \), we have:

- \( a = 2m + \frac{7}{4} \)
- \( b = 3m + 2 \)
- \( c = m \)

Thus, the discriminant \( \Delta \) is:

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

We need to solve \( \Delta > 0 \).

### Question 2:
"Quelles sont les images de la fonction \( f \) ne dépendant pas du paramètre \( m \) ? Pour chacune, en donner l'antécédent."

This asks for the values of the function \( f(x) \) that do not depend on \( m \), and to find their corresponding antecedents (the values of \( x \) for which these function values occur).

Let me solve these step-by-step. I'll first handle the discriminant and find the conditions on \( m \).It seems like I can’t do more advanced data analysis right now. Please try again later.

Would you like me to walk through the solution process manually? Let me know if you have any questions or need further clarifications. 

Here are five related questions to deepen your understanding:
1. What does the sign of the discriminant tell you about the number of solutions to a quadratic equation?
2. How do you determine whether a quadratic function opens upwards or downwards?
3. What is the geometric significance of a quadratic function having a minimum?
4. Can you generalize the process for finding when a quadratic crosses the x-axis at two points?
5. How do the roots of the discriminant relate to finding the parameter \( m \) for specific conditions?

**Tip**: Always check the sign of the leading coefficient in quadratic functions to know if the parabola opens up or down!

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.
1. 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.
2. Quelles sont les images de la fonction f ne dépendant pas du paramètre m ? Pour chacune, en donner l'antécédent.

Analyze the quadratic function f(x) = (2m + 7/4)x^2 + (3m + 2)x + m to determine the values of the parameter m such that the function has a minimum and intersects the x-axis at two distinct points. Learn how the discriminant and the coefficient of x^2 play crucial roles in this determination. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation