(1) (i) Suppose that Ax + Bx+C 0. Show that B 4AC. (ii) Prove...

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(1) (i) Suppose that Ax² + Bx+C ≥ 0. Show that B² ≤ 4AC. (ii) Prove that is a₁, ..., an, b₁, ..., b, are any real numbers then (a₁b₁ +...+anbn)² ≤ (a² + ... + a²) (b² + ... + b ²2). This is the Cauchy-Schwartz inequality. Hint: Consider an expression of the form (a₁x + b₁)²+...+(anx + bn) ². (iii) Let a₁, ..., an, be positive real numbers. Let An denote the arithmetic mean and Hn the harmonic mean. ie. An 1-))-¹. Prove that an 1-1 ak and H₂ k=1 = ( / (+ + ... n Hn ≤ An. Hint: Use (ii). + = (1) (i) Suppose that Ax² + Bx+C ≥ 0. Show that B² ≤ 4AC. (ii) Prove that is a₁, ..., an, b₁, ..., b, are any real numbers then (a₁b₁ +...+anbn)² ≤ (a² + ... + a²) (b² + ... + b ²2). This is the Cauchy-Schwartz inequality. Hint: Consider an expression of the form (a₁x + b₁)²+...+(anx + bn) ². (iii) Let a₁, ..., an, be positive real numbers. Let An denote the arithmetic mean and Hn the harmonic mean. ie. An 1-))-¹. Prove that an 1-1 ak and H₂ k=1 = ( / (+ + ... n Hn ≤ An. Hint: Use (ii). + =