(a) Let f(x) = px + 2qx +r with p > 0. By considering the minimum,...

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(a) Let f(x) = px² + 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,bı + azb2 + + anbn, and also the norms of these vectors are defined as |lā|| = Vā · ā = a+ a3 + .+ an, |||| b} + b3 + · …. + b%. %3D Prove the Cauchy-Schwarz inequality (a · b)? < |lä||2||b||?, that is the inequality (ajbı + azb2 + ...+ anbn)² < (a² + a² + + an)(bỉ + b3 +.. + b). Hint: Consider the function f(x) = (a1x+ b1)² + (a2x + b2)² + ·+ (anx+ bn )² and apply as'u Etkinl (a) Let f(x) = px² + 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,bı + azb2 + + anbn, and also the norms of these vectors are defined as |lā|| = Vā · ā = a+ a3 + .+ an, |||| b} + b3 + · …. + b%. %3D Prove the Cauchy-Schwarz inequality (a · b)? < |lä||2||b||?, that is the inequality (ajbı + azb2 + ...+ anbn)² < (a² + a² + + an)(bỉ + b3 +.. + b). Hint: Consider the function f(x) = (a1x+ b1)² + (a2x + b2)² + ·+ (anx+ bn )² and apply as'u Etkinl