Mathematics

Problem

For a real parameter

k

, consider the quartic polynomial

F_k(x)=x^{4}-4kx^{2}+k^{2}

. a. (i) Show that the critical points of

F_k(x)

are the real solutions of

x^{3}-2kx=0

. b. (ii) Determine all real values of

k

for which

F_k(x)

possesses exactly two distinct local minima on the real line. c. (iii) For the values of

k

obtained in part (ii), compute the coordinates of these local minima and the corresponding minimum values of

F_k(x)