Inequalities

\(a,b,c,d>0\)

  1. \(a+\frac{1}{a}\geq?\)
  2. \(\frac{a+b}{2}(\geq or\leq?)\sqrt{ab}\)
  3. \(\frac{a+b}{2}(\geq or\leq?)\sqrt{ab}\) (another version)
  4. \(2ab(\geq or\leq?)a^{2}+b^{2}\)
  5. \(ab+bc+ca(\geq or\leq?)a^{2}+b^{2}+c^{2}\)
  6. \(\frac{a}{b}<\frac{c}{d}\Rightarrow\frac{a}{b}<?<\frac{c}{d}\)
  7. Cauchy-Schwarz inequality \(|px+qy|\leq?\)