(Long Answer – Extended Problem) A function is defined by h(x)= cases x^ 2 , & x 0,…

MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation

Problem

(Long Answer – Extended Problem) A function

h

is defined by

h(x)=\begin{cases} x^{2}, & x\le 0,\\ |x|, & 0<x\le 2,\\ 4-(x-2)^{2}, & x>2. \end{cases}

(i) Find

h'(x)

on each sub-interval where it exists. (ii) List all critical points of

h

in the interval

[-2,4]

. (iii) For every critical point obtained in part (ii), determine whether

h

attains a local maximum, a local minimum or neither. Justify your answer by analysing the sign of

h'(x)

on the appropriate intervals. (iv) Hence, specify the complete set of intervals where

h

is increasing and where it is decreasing. (v) Sketch a rough graph of

y=h(x)

, clearly indicating the critical points and the nature of each.