For a real parameter

MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation

Problem

k

, consider the quartic polynomial

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

. (i) Show that the critical points of

F_k(x)

are the real solutions of

x^{3}-2kx=0

. (ii) Determine all real values of

k

for which

F_k(x)

possesses exactly two distinct local minima on the real line. (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)