Let and be twice-differentiable functions on an interval with and
MATHEMATICS · Multi-part · CBSE · Mathematics · Continuity and Differentiability · Continuity and Differentiability
Problem
Let p(t) and q(t) be twice-differentiable functions on an interval a<t<b with dtdx=p(t) and dtdy=q(t). Suppose p(t) changes sign at t=c with a<c<b and p(c)=0, while q′(t) is known to be positive on the entire interval.
(a) Explain why the potential change in sign of dx2d2y at t=c cannot be decided solely from the sign of q′(t).
(b) Describe the additional information about p(t) or its derivatives that must be examined to reach a definitive conclusion about the concavity of the curve on intervals immediately to the left and right of t=c.
(c) Provide a coherent logical argument—without specific numerical computation—that outlines how one would decide whether an inflection point occurs at t=c.