Mathematics

Problem

Let

f(x)=x^{3}-3kx^{2}+3(k^{2}-1)x+7

, where

k

is a real parameter. a. (i) Find the critical points of

f

in terms of

k

. b. (ii) Obtain the condition on

k

for which

f

possesses exactly two distinct real critical points. c. (iii) For those values of

k

obtained in part (ii), calculate the corresponding local maximum value

M

and local minimum value

m

f

. d. (iv) Determine all real values of

k

for which

M-m\ge 8