Mathematics

Problem

Let

k

be a real parameter and consider the function

f_k(x)=x^{3}-3k x^{2}+4

, defined for all real

x

. a. (i) Obtain the critical points of

f_k(x)

in terms of

k

. b. (ii) For each critical point found in part (i), determine the nature of the extremum (local maximum or local minimum) in terms of

k

. c. (iii) Find the set of all real values of

k

for which

f_k(x)

is strictly increasing on

(-\infty,\infty)