The concept of a critical point is central to analyzing the local behavior of…
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
The concept of a critical point is central to analyzing the local behavior of differentiable functions.
Answer the following questions about critical points in the context of polynomial functions.
(a) Define a critical point of a differentiable function f and describe how a change in sign of f′(x) in a neighborhood of that point is related to the existence of a local extremum.
(b) Explain why, for any polynomial p(x), the only candidates for critical points are the real numbers that satisfy p′(x)=0.
(c) Let p(x)=(x−2)3(x+1)2. Without explicitly computing p′(x), predict the number of distinct critical points of p and describe the expected qualitative behavior of p near each critical point. Justify your conclusions in terms of the multiplicities of the factors of p(x).