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4. Set p(x) = x³ + ax²+bx+c. a) Show that the graph of p has exactly one point of inflection. What is x at that point? b) Show that p has two local extreme values if and only if a² > 3b. c) Show that p cannot have only one local extreme value. d) Show that if p has a local maximum and a local minimum, then the midpoint of the line segment that connects the local high point to the local low point is a point of inflection.

4. Set p(x) = x³ + ax²+bx+c. a) Show that the graph of p has exactly one point of inflection. What is x at that point? b) Show that p has two local extreme values if and only if a² > 3b. c) Show that p cannot have only one local extreme value. d) Show that if p has a local maximum and a local minimum, then the midpoint of the line segment that connects the local high point to the local low point is a point of inflection.

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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4. Set p(x) = x³ + ax²+bx+c. a) Show that the graph of p has exactly one point of inflection. What is x at that point? b) Show that P has two local extreme values if and only if a² > 3b. c) Show that p cannot have only one local extreme value. d) Show that if p has a local maximum and a local minimum, then the midpoint of the line segment that connects the local high point to the local low point is a point of inflection.
Transcribed Image Text:4. Set p(x) = x³ + ax²+bx+c. a) Show that the graph of p has exactly one point of inflection. What is x at that point? b) Show that P has two local extreme values if and only if a² > 3b. c) Show that p cannot have only one local extreme value. d) Show that if p has a local maximum and a local minimum, then the midpoint of the line segment that connects the local high point to the local low point is a point of inflection.
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