**Determining Intervals of Concavity and Inflection Points****Problem Statement:**Suppose that the second derivative of a function \( f \) is \[ f''(x) = -(x + 1)^3 (x - 1)^2 (x - 4)^5. \]Determine the intervals of concavity and the \( x \)-values of any inflection points.**Solution Procedure:**1. **Identify Critical Points:** To find the intervals of concavity and inflection points, first identify the critical points where \( f''(x) = 0 \). \[ -(x + 1)^3 (x - 1)^2 (x - 4)^5 = 0 \] Solving the above equation, we get the critical points: \[ x = -1, x = 1, x = 4 \]2. **Test Intervals:** Analyze the sign of \( f''(x) \) in the intervals determined by the critical points: - \( (-\infty, -1) \) - \( (-1, 1) \) - \( (1, 4) \) - \( (4, \infty) \)3. **Concavity and Inflection Points:** - For \( x < -1 \): Choose a test point, say \( x = -2 \). Evaluating \( f''(-2) \) will determine the sign of the second derivative in this interval. - For \( -1 < x < 1 \): Choose a test point, say \( x = 0 \). Evaluating \( f''(0) \) will determine the sign of the second derivative in this interval. - For \( 1 < x < 4 \): Choose a test point, say \( x = 2 \). Evaluating \( f''(2) \) will determine the sign of the second derivative in this interval. - For \( x > 4 \): Choose a test point, say \( x = 5 \). Evaluating \( f''(5) \) will determine the sign of the second derivative in this interval. By analyzing the sign changes of \( f''(x) \) in these intervals, you can determine the concavity and identify any inflection points where

Name: **Determining Intervals of Concavity and Inflection Points****Problem
Uploaded: 2024-03-20T06:47:22.000Z
Channel: Bartleby
Description: **Determining Intervals of Concavity and Inflection Points****Problem Statement:**Suppose that the second derivative of a function \( f \) is \[ f''(x) = -(x + 1)^3 (x - 1)^2 (x - 4)^5. \]Determine the intervals of concavity and the \( x \)-values o