f(x)=x/(x^2+8x+15)
a) Give the domain of f
 (in interval notation) 

b) Find the critical numbers of f
. 
 (Separate multiple answers by commas.)

c) Determine the intervals on which f
 is increasing and decreasing.
f
 is increasing on: 

f
 is decreasing on: 

d) Use the First Derivative Test to determine whether each critical point is a relative maximum, minimum, or neither.
Relative maxima occur at x=
 
 (Separate multiple answers by commas.)
Relative minima occur at x=
 
 (Separate multiple answers by commas.)

To analyze the function \( f(x) = \frac{x}{x^2 + 8x + 15} \), let's break down each part:

### a) Domain of \( f \)

The function is defined wherever the denominator is not zero. We need to find the values of \( x \) that make the denominator zero:

\[
x^2 + 8x + 15 = 0
\]

Factoring gives:

\[
(x + 3)(x + 5) = 0
\]

Thus, \( x = -3 \) and \( x = -5 \). The domain of \( f \) in interval notation is:

\[
\text{Domain: } (-\infty, -5) \cup (-5, -3) \cup (-3, \infty)
\]

### b) Critical Numbers of \( f \)

To find the critical numbers, we first find the derivative of \( f \) using the quotient rule:

\[
f'(x) = \frac{(x^2 + 8x + 15)(1) - x(2x + 8)}{(x^2 + 8x + 15)^2}
\]

Simplifying the numerator:

\[
= (x^2 + 8x + 15) - (2x^2 + 8x) = -x^2 + 15
\]

Setting the derivative equal to zero to find critical points:

\[
-x^2 + 15 = 0 \implies x^2 = 15 \implies x = \pm \sqrt{15}
\]

Thus, the critical numbers are:

\[
\text{Critical numbers: } \sqrt{15}, -\sqrt{15}
\]

### c) Intervals of Increase and Decrease

To determine where \( f \) is increasing or decreasing, we analyze the sign of \( f' \):

1. **Intervals to test**: Choose points in the intervals \( (-\infty, -\sqrt{15}) \), \( (-\sqrt{15}, -5) \), \( (-5, -3) \), \( (-3, \sqrt{15}) \), \( (\sqrt{15}, \infty) \).
2. Evaluate the sign of \( f' \):

- For \( x < -\sqrt{15} \) (e.g., \( -4 \)): \( f'(-4) > 0 \) (increasing)
- For \( -\sqrt{15} < x < -5 \) (e.g., \( -6 \)): \( f'(-6) < 0 \) (decreasing)
- For \( -5 < x < -3 \): \( f' \) is not defined (as \( f \) has vertical asymptotes).
- For \( -3 < x < \sqrt{15} \): \( f' > 0 \) (increasing)
- For \( x > \sqrt{15} \): \( f' < 0 \) (decreasing)

Thus:

- \( f \) is increasing on \( (-\infty, -\sqrt{15}) \) and \( (-3, \sqrt{15}) \).
- \( f \) is decreasing on \( (-\sqrt{15}, -5) \) and \( (\sqrt{15}, \infty) \).

### d) First Derivative Test

To determine whether each critical point is a relative maximum, minimum, or neither, we examine the sign of \( f' \) around each critical number:

- **At \( x = -\sqrt{15} \)**:
  - \( f' \) changes from positive to negative (increasing to decreasing): **Relative maximum**.

- **At \( x = \sqrt{15} \)**:
  - \( f' \) changes from negative to positive (decreasing to increasing): **Relative minimum**.

### Summary

1. **Domain**: \( (-\infty, -5) \cup (-5, -3) \cup (-3, \infty) \)
2. **Critical Numbers**: \( \sqrt{15}, -\sqrt{15} \)
3. **Increasing on**: \( (-\infty, -\sqrt{15}) \) and \( (-3, \sqrt{15}) \)
4. **Decreasing on**: \( (-\sqrt{15}, -5) \) and \( (\sqrt{15}, \infty) \)
5. **Relative maxima at**: \( x = -\sqrt{15} \)
6. **Relative minima at**: \( x = \sqrt{15} \)

Do you have any questions or need further details? Here are some related questions you might consider:

1. How do you derive the quotient rule?
2. What are the implications of vertical asymptotes on the behavior of the function?
3. Can you explain the significance of critical points in optimization problems?
4. What happens to the function behavior near the critical points?
5. How would you find inflection points for this function? 

**Tip**: Always sketch the graph to visualize the behavior of the function around critical points and asymptotes.

f(x)=x/(x^2+8x+15) a) Give the domain of f (in interval notation) b) Find the critical numbers of f. c) Determine the intervals on which f is increasing and decreasing. d) Use the First Derivative Test to determine whether each critical point is a relative maximum, minimum, or neither.

Explore the analysis of the function f(x)=x/(x^2+8x+15). This guide covers finding the domain, critical numbers, intervals of increase and decrease, and applying the First Derivative Test for identifying relative maxima and minima. Perfect for advanced calculus students.

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