Consider the function f(x)=3x+94x+3f(x)=3x+94x+3. For this function there are two important intervals: (−∞,A)(-∞,A) and (A,∞)(A,∞) where the function is not defined at AA.
Find AA   

For each of the following intervals, tell whether f(x)f(x) is increasing or decreasing.
(−∞,A)(-∞,A): Select an answer Increasing Decreasing 
(A,∞)(A,∞) Select an answer Increasing Decreasing 

Note that this function has no inflection points, but we can still consider its concavity. For each of the following intervals, tell whether f(x)f(x) is concave up or concave down.
(−∞,A)(-∞,A): Select an answer Concave Up Concave Down 
(A,∞)(A,∞) Select an answer Concave Up Concave Down 

Let's solve each part of the problem step-by-step. 

### 1. Determine \( A \) where the function is undefined

The function given is:
\[
f(x) = \frac{3x + 9}{4x + 3}
\]

The function is undefined where the denominator equals zero. Thus, we solve:
\[
4x + 3 = 0
\]
\[
4x = -3
\]
\[
x = -\frac{3}{4}
\]

So, \( A = -\frac{3}{4} \).

### 2. Determine whether \( f(x) \) is increasing or decreasing on each interval

To determine whether \( f(x) \) is increasing or decreasing on the intervals \((-\infty, A)\) and \((A, \infty)\), we find the derivative \( f'(x) \) and examine its sign.

#### Finding \( f'(x) \)
Using the quotient rule for derivatives, where \( f(x) = \frac{u}{v} \) with \( u = 3x + 9 \) and \( v = 4x + 3 \):
\[
f'(x) = \frac{v \cdot u' - u \cdot v'}{v^2}
\]
- \( u' = 3 \)
- \( v' = 4 \)

Thus,
\[
f'(x) = \frac{(4x + 3)(3) - (3x + 9)(4)}{(4x + 3)^2}
\]
Simplifying,
\[
f'(x) = \frac{12x + 9 - 12x - 36}{(4x + 3)^2}
\]
\[
f'(x) = \frac{-27}{(4x + 3)^2}
\]

Since \((4x + 3)^2\) is always positive, \( f'(x) \) will always be negative (because the numerator \(-27\) is negative). This implies that \( f(x) \) is always **decreasing** on both intervals.

- On \((-\infty, A)\): **Decreasing**
- On \((A, \infty)\): **Decreasing**

### 3. Determine the concavity of \( f(x) \) on each interval

To determine concavity, we need the second derivative \( f''(x) \) and its sign.

#### Finding \( f''(x) \)
To differentiate \( f'(x) = \frac{-27}{(4x + 3)^2} \) again, we can use the chain rule:
\[
f''(x) = \frac{d}{dx} \left( -27 \cdot (4x + 3)^{-2} \right)
\]
Letting \( g(x) = (4x + 3)^{-2} \), we find \( g'(x) \):
\[
g'(x) = -2(4x + 3)^{-3} \cdot 4 = -8(4x + 3)^{-3}
\]
Thus,
\[
f''(x) = -27 \cdot \left(-8(4x + 3)^{-3}\right) = 216(4x + 3)^{-3}
\]

Since \((4x + 3)^{-3}\) is positive for \( x > -\frac{3}{4} \) and negative for \( x < -\frac{3}{4} \), we find:
- On \((-\infty, A)\): **Concave Down**
- On \((A, \infty)\): **Concave Up**

### Summary

- **Value of \( A \):** \( A = -\frac{3}{4} \)
- **Monotonicity:**
  - \((-\infty, A)\): Decreasing
  - \((A, \infty)\): Decreasing
- **Concavity:**
  - \((-\infty, A)\): Concave Down
  - \((A, \infty)\): Concave Up

Would you like a deeper explanation on any step, or do you have questions?

#### Related Questions:
1. What is the significance of a function's concavity in understanding its graph?
2. How can we interpret the behavior of a function around a vertical asymptote?
3. Why is it helpful to know where a function is increasing or decreasing?
4. What is the importance of finding intervals of concavity for real-world applications?
5. How does the derivative inform us about the function's slope and behavior?

#### Tip:
When analyzing rational functions, always start by identifying points of discontinuity (where the denominator is zero) to understand the function's domain.

Consider the function f(x) = (3x + 9) / (4x + 3). For this function there are two important intervals: (-∞, A) and (A, ∞) where the function is not defined at A. Find A. For each of the following intervals, tell whether f(x) is increasing or decreasing. (-∞, A): Select an answer Increasing Decreasing (A, ∞) Select an answer Increasing Decreasing. Note that this function has no inflection points, but we can still consider its concavity. For each of the following intervals, tell whether f(x) is concave up or concave down. (-∞, A): Select an answer Concave Up Concave Down (A, ∞) Select an answer Concave Up Concave Down.

Explore how to find intervals of increase, decrease, and concavity for the rational function f(x) = (3x + 9) / (4x + 3). This solution includes key calculus concepts such as the quotient rule, first and second derivatives, and concavity analysis. Suitable for advanced high school students.

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