Graph the​ function, considering the​ domain, critical​ points, symmetry, relative​ extrema, regions where the function is increasing or​ decreasing, inflection​ points, regions where the function is concave upward or concave​ downward, intercepts where​ possible, and asymptotes where applicable.
f left parenthesis x right parenthesis equalsStartFraction negative 4 x plus 8 Over 4 x plus 3 EndFraction
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Part 1
Find the first and second derivatives of​ f(x).
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Let's work through the problem step by step.

Given:  
\[
f(x) = \frac{-4x + 8}{4x + 3}
\]

We will first find the **first derivative** \( f'(x) \) and the **second derivative** \( f''(x) \).

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### **1. First Derivative \( f'(x) \)**

We use the **quotient rule** for derivatives:
\[
\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}
\]
where \( u = -4x + 8 \) and \( v = 4x + 3 \).

#### Step 1: Compute \( u' \) and \( v' \)
- \( u = -4x + 8 \quad \Rightarrow \quad u' = -4 \)
- \( v = 4x + 3 \quad \Rightarrow \quad v' = 4 \)

#### Step 2: Apply the quotient rule
\[
f'(x) = \frac{(-4)(4x + 3) - (-4x + 8)(4)}{(4x + 3)^2}
\]

#### Step 3: Expand the numerator
\[
(-4)(4x + 3) = -16x - 12
\]
\[
(-4x + 8)(4) = -16x + 32
\]
Thus:
\[
f'(x) = \frac{-16x - 12 + 16x - 32}{(4x + 3)^2}
\]

#### Step 4: Simplify the numerator
\[
f'(x) = \frac{-44}{(4x + 3)^2}
\]

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### **2. Second Derivative \( f''(x) \)**

We now differentiate the first derivative \( f'(x) = \frac{-44}{(4x + 3)^2} \) using the **quotient rule** and chain rule.

First, rewrite the derivative:
\[
f'(x) = -44 \cdot (4x + 3)^{-2}
\]

Now apply the **chain rule**:
\[
f''(x) = -44 \cdot (-2) \cdot (4x + 3)^{-3} \cdot 4
\]

Simplify:
\[
f''(x) = \frac{352}{(4x + 3)^3}
\]

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### **Summary**

- **First derivative:**
  \[
  f'(x) = \frac{-44}{(4x + 3)^2}
  \]

- **Second derivative:**
  \[
  f''(x) = \frac{352}{(4x + 3)^3}
  \]

Would you like to proceed with analyzing the graph, such as finding the intercepts, critical points, and concavity regions?

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### 5 Related Questions:
1. How do we find the critical points using the first derivative?
2. What does the second derivative tell us about concavity and inflection points?
3. How do we determine the vertical and horizontal asymptotes of this function?
4. What is the domain of \( f(x) = \frac{-4x + 8}{4x + 3} \)?
5. How do we determine where the function is increasing or decreasing?

**Tip:** Always double-check your signs when applying the quotient and chain rules to avoid mistakes.

Graph the function, considering the domain, critical points, symmetry, relative extrema, regions where the function is increasing or decreasing, inflection points, regions where the function is concave upward or concave downward, intercepts where possible, and asymptotes where applicable. f(x) = (-4x + 8) / (4x + 3). Part 1: Find the first and second derivatives of f(x).

Explore the graph of the rational function f(x) = (-4x + 8) / (4x + 3) by finding its domain, critical points, asymptotes, and analyzing its concavity and extrema. This solution includes step-by-step derivative calculations and interpretation for a complete graphing analysis, ideal for advanced calculus students.

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