MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions
Problem
Three functions are defined analytically by
p(x)=e−x2,
q(x)=−2xe−x2,
r(x)=(4x2−2)e−x2.
The graphs of these functions are plotted together but not labeled. Use the information below to answer the questions that follow.
(a) Identify which of p, q, and r serves as the original function, which as its first derivative, and which as its second derivative. State your reasoning briefly.
(b) Confirm your identification in part (a) by computing p′(x) and q′(x), showing all work.
(c) Using your results, find all x–coordinates of inflection points of the original function, and justify your answer.