Q- if A^2+B^2+C^2=1, then what is the range of AB+BC+CA

sol:-
A+B+C>=0

(A+B+C)^2>=0

A^2+B^+C^2+2(AB+BC+CA)>=0

1+2(AB+BC+CA)>=0

AB+BC+CA>= -1/2
Now,
(A-B)^2>=0
(B-C)^2>=0
(C-A)^2>=0

(A-B)^2+(B-C)^2+(C-A)^2>=0

2(A^2+B^2+C^2)-2(AB+BC+CA)>=0

(A^2+B^2+C^2)-(AB+BC+CA)>=0

1-(AB+BC+CA)>=0

-(AB+BC+CA)>= -1

(AB+BC+CA)<=1

Range of  AB+BC+CA=[-1/2,1].