(Long Answer – Extended Problem)
A function h is defined by
h(x)=⎩⎨⎧x2,∣x∣,4−(x−2)2,x≤0,0<x≤2,x>2.
(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.