Mathematics

Problem

For a real parameter

k

, define the quadratic function

F_k:\mathbb R\to\mathbb R

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

. a. (i) Express

F_k(x)

in completed-square form. b. (ii) Using the form obtained in part (i), find the least value of

F_k(x)

in terms of

k

. c. (iii) Hence determine the range of

F_k

for the following cases: (a)

k>-1

, (b)

k=-1

, (c)

k<-1