A) Given f(x) = ax^2 + 2bx + c with a>0. By considering the minimum, prove that f(x)>=0 for all real x if and only if b^2-ac<=0.
B) From A, let f(x)= (a1x+b1)^2 + (a2x+b2)^2 + ...... +(anx + bn)^2 and deduce Schwarz's inequality: (a1b1 + a2b2 + .... +anbn)^2 <= (a1^2 + a2^2 + ..... + an^2)(b1^2 + b2^2 + ..... + bn^2)
C) Show that equality holds in Schwarz's inequality only if there exists a real number x that makes aix equal to -bi for every value of i from 1 to n. (the i's are subs)