Mathematics

Problem

Consider the cubic polynomial

f(x)=x^{3}-3k x^{2}+3(k^{2}-1)x+5,

where

k

is a real parameter. (i) Obtain all critical points of

f

in terms of

k

. (ii) Determine the nature (local maximum or local minimum, if any) of each critical point. (iii) Hence find the set of all real values of

k

for which the curve

y=f(x)

possesses exactly one local minimum but no local maximum.