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Find a cubic tunchion feo=ax34 bx'textd that has a local maximum valve of 3 at X=-2 and a local minimum value of o at x = l

Find a cubic tunchion feo=ax34 bx'textd that has a local maximum valve of 3 at X=-2 and a local minimum value of o at x = l

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Problem Statement:** Find a cubic function \( f(x) = ax^3 + bx^2 + cx + d \) that has a local maximum value of 3 at \( x = -2 \) and a local minimum value of 0 at \( x = 1 \). **Explanation:** To solve this problem, consider the following steps: 1. **Local Maximum and Minimum Conditions:** - A local maximum or minimum occurs where the first derivative \( f'(x) \) is equal to zero. - The second derivative test can confirm if those points are maxima or minima. 2. **Derivatives:** - First derivative: \( f'(x) = 3ax^2 + 2bx + c \). - Second derivative: \( f''(x) = 6ax + 2b \). 3. **Use Given Conditions:** - At \( x = -2 \), \( f(-2) = 3 \) (local maximum). - At \( x = 1 \), \( f(1) = 0 \) (local minimum). 4. **Set Up Equations:** - Use the derivative and condition to solve for the coefficients \( a, b, c, \) and \( d \). By analyzing and solving these equations, you can determine the specific values of \( a, b, c, \) and \( d \) for the function described.
Transcribed Image Text:**Problem Statement:** Find a cubic function \( f(x) = ax^3 + bx^2 + cx + d \) that has a local maximum value of 3 at \( x = -2 \) and a local minimum value of 0 at \( x = 1 \). **Explanation:** To solve this problem, consider the following steps: 1. **Local Maximum and Minimum Conditions:** - A local maximum or minimum occurs where the first derivative \( f'(x) \) is equal to zero. - The second derivative test can confirm if those points are maxima or minima. 2. **Derivatives:** - First derivative: \( f'(x) = 3ax^2 + 2bx + c \). - Second derivative: \( f''(x) = 6ax + 2b \). 3. **Use Given Conditions:** - At \( x = -2 \), \( f(-2) = 3 \) (local maximum). - At \( x = 1 \), \( f(1) = 0 \) (local minimum). 4. **Set Up Equations:** - Use the derivative and condition to solve for the coefficients \( a, b, c, \) and \( d \). By analyzing and solving these equations, you can determine the specific values of \( a, b, c, \) and \( d \) for the function described.
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