(a) Let f(x) = px + 2qx +r with p > 0. By considering the minimum, prove that f(x) > 0 for all real x if and only if q? pr < 0. (b) Let a = (a1, a2, ..., an) and b = (b1, b2, ..., bn) be any two vectors in R". The inner product (dot product) of these two vectors are defined as a b = a,b + azb2 + + anbn, and also the norms of these vectors are defined as |l|| = V = a+ a3 + .+ an, |||| b} + b3 + . + b%. %3D Prove the Cauchy-Schwarz inequality (a b)? < |l||2||b||?, that is the inequality (ajb + azb2 + ...+ anbn) < (a + a + + an)(b + b3 +.. + b). Hint: Consider the function f(x) = (a1x+ b1) + (a2x + b2) + + (anx+ bn ) and apply as'u Etkinl