Use the first derivative test and the second derivative test to determine where each function is increasing, decreasing, concave up, and concave down. 6 y = - -, x# - 5 (5 + x)?
Use the first derivative test and the second derivative test to determine where each function is increasing, decreasing, concave up, and concave down. 6 y = - -, x# - 5 (5 + x)?
Chapter3: Functions
Section3.5: Transformation Of Functions
Problem 5SE: How can you determine whether a function is odd or even from the formula of the function?
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![**Topic: First and Second Derivative Tests**
**Objective:**
Use the first derivative test and the second derivative test to determine where the function is increasing, decreasing, concave up, and concave down.
**Function:**
\[ y = \frac{6}{(5 + x)^2}, \quad x \neq -5 \]
**Instructions:**
1. **First Derivative Test**: Find \( y' \) to identify the intervals where the function is increasing or decreasing.
2. **Second Derivative Test**: Find \( y'' \) to determine the intervals of concavity, specifically where the function is concave up or concave down.
**Approach:**
- Differentiate the function with respect to \( x \) to obtain the first derivative, \( y' \), and analyze sign changes.
- Find the second derivative, \( y'' \), to assess concavity.
- Determine critical points and inflection points based on these derivatives to conclude the behavior of the function over different intervals.
**Note:**
The point \( x = -5 \) is excluded from the domain of the function as it would result in division by zero.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb4e8e2a5-a7b3-45d7-b3fc-bbdb74e12e7d%2Fa3030908-3318-4d5b-abe6-234269ce203c%2Fqbywsj_processed.png&w=3840&q=75)
Transcribed Image Text:**Topic: First and Second Derivative Tests**
**Objective:**
Use the first derivative test and the second derivative test to determine where the function is increasing, decreasing, concave up, and concave down.
**Function:**
\[ y = \frac{6}{(5 + x)^2}, \quad x \neq -5 \]
**Instructions:**
1. **First Derivative Test**: Find \( y' \) to identify the intervals where the function is increasing or decreasing.
2. **Second Derivative Test**: Find \( y'' \) to determine the intervals of concavity, specifically where the function is concave up or concave down.
**Approach:**
- Differentiate the function with respect to \( x \) to obtain the first derivative, \( y' \), and analyze sign changes.
- Find the second derivative, \( y'' \), to assess concavity.
- Determine critical points and inflection points based on these derivatives to conclude the behavior of the function over different intervals.
**Note:**
The point \( x = -5 \) is excluded from the domain of the function as it would result in division by zero.
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