(Long Answer – Comprehensive Analysis)
Define
f(x)=∣x3−3x∣,x∈R.
(i) Express f(x) as a piece-wise function without the absolute value sign.
(ii) Compute f′(x) on each interval where the expression in part (i) is differentiable.
(iii) Find all critical points of f.
(iv) For each critical point, analyse the sign of f′(x) on the neighbouring intervals and classify the point as a local maximum, a local minimum or neither.
(v) Determine the intervals of increase and decrease of f on [−3,3].
(vi) Draw a neat labelled sketch of y=f(x) on [−3,3], indicating clearly the nature of each critical point.