Consider the function g(x)=x1/3(x−2)2 defined for 0≤x≤4.
(i) Obtain g′(x) and list all x in [0,4] where g′(x)=0 or g′(x) does not exist but g(x) is continuous.
(ii) Compute g′′(x) wherever it exists and apply it to classify the critical points identified in part (i), indicating clearly any point where the second derivative test is not applicable or inconclusive.
(iii) For each point where the second derivative test failed in part (ii), analyse the sign of g′(x) on intervals immediately to the left and right of the point to decide whether it is a local maximum, local minimum, or neither.
(iv) Comment on the behaviour of g at the endpoints x=0 and x=4 in comparison with the local extrema found.