Mathematics

Problem

Let

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

where

k

is a positive real number. a. (i) Show that

g'(x)=4x^{3}+8kx

has at most one real zero. b. (ii) Hence, prove that

g

has exactly one local minimum and no local maximum for every

k>0