Mathematics

Problem

For the family of functions

F_k(x)=x^{3}-3k x+7,

k

being a real parameter, consider the behaviour of stationary points. a. (i) Determine the coordinates of all stationary points in terms of

k

. b. (ii) Find the set of all real values of

k

for which the function possesses both a local maximum and a local minimum. c. (iii) For those values of

k

obtained in part (ii), identify which stationary point corresponds to the maximum and which to the minimum.