A function is defined by the piecewise rule p(x)= cases 9-(x+1)^2 , & -4 x -1, |x+2|,…

MATHEMATICS · Multi-part · CBSE · Mathematics · Continuity and Differentiability · Continuity and Differentiability

Problem

A function

p

is defined by the piecewise rule

p(x)=\begin{cases} \sqrt{9-(x+1)^2}, & -4\le x\le -1,\\ |x+2|, & -1<x<3,\\ \dfrac{1}{2}(x-3)^2, & x\ge 3. \end{cases}

Assume that

p

is extended smoothly on each piece as indicated and that the domain of

p

[-4,\infty)

. (1) Part 1: (a) Determine all

x

–values in the domain of

p

at which

p

is not differentiable. For each such value, state the specific reason the derivative fails to exist. (2) Part 2: (b) For each interval on which

p

is differentiable, obtain an explicit formula for

p'(x)

. (3) Part 3: (c) Sketch a complete graph of

p'

, labeling any discontinuities, cusp points, or corners. Indicate clearly the one–sided limits of

p'

where they differ. (4) Part 4: (d) Provide a justification that the sign of

p'

on each interval you identified is consistent with the qualitative behavior (increasing/decreasing) of

p

on that interval.