The given function is:

\[
f(x) = \frac{6x}{\sqrt{9 - 4x^2}}
\]

This function represents a rational function involving a square root in the denominator. To analyze it, we can look at several aspects:

### 1. **Domain of the function**
To find the domain, we need to determine when the square root is defined, which means the expression under the square root must be non-negative. In other words, we need:

\[
9 - 4x^2 \geq 0
\]

Let's solve this inequality:
\[
9 \geq 4x^2
\]
\[
\frac{9}{4} \geq x^2
\]
\[
|x| \leq \frac{3}{2}
\]

Thus, \( x \) must lie within the interval \( \left[ -\frac{3}{2}, \frac{3}{2} \right] \). Therefore, the domain is:

\[
x \in \left[ -\frac{3}{2}, \frac{3}{2} \right]
\]

### 2. **Asymptotes and behavior**
- **Vertical asymptotes:** Since the denominator involves a square root, the function will tend to infinity as \( x \) approaches the endpoints of the domain, \( x = \pm \frac{3}{2} \), where the denominator becomes zero.

### 3. **Differentiation**
To find the derivative of \( f(x) \), we can use the quotient rule, given that the function is a quotient of two functions. The quotient rule states:

\[
f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2}
\]

Would you like to proceed with this calculation? Or would you prefer more details on specific parts? Let me know.

---

Here are 5 additional questions to consider:

1. What is the behavior of the function at the boundaries of its domain?
2. How does the function behave as \( x \to 0 \)?
3. Can the function be simplified in any way?
4. What does the graph of this function look like?
5. Are there any critical points (e.g., maximum or minimum)?

### Tip:
When differentiating complex functions like this one, the quotient and chain rules are commonly used to find the derivative effectively.

Explore the domain, asymptotes, and differentiation of the function f(x) = 6x / √(9 - 4x^2). This problem involves key calculus concepts, including rational functions, domain analysis, and the quotient rule for differentiation. Suitable for students in Grades 11-12 or early college levels.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation