Introduction

Here is a version of Tutorial Question 4 off of Problem Sheet 3 with an alternative solution for part c). Parts a) and b) are included for completeness.

Example 1: (PS3 Question 4)
  1. Show that \[ 2xy \leq x^2 + y^2, \;\; \forall x,y \in \mathbb{R},\] and that equality holds only if \(x = y\).
  2. Show that \[\sqrt{\frac{x}{2}} + \sqrt{\frac{y}{2}} \leq \sqrt{x + y} \leq \sqrt{x} + \sqrt{y}, \;\; \forall x,y > 0.\]
  3. Prove that \[\lvert \sqrt{1 + x^2} - \sqrt{1 + y^2} \rvert \leq \lvert x - y \rvert \;\; \forall x,y \in \mathbb{R}.\]