Prove the Arithmetic Mean–Geometric Mean inequality: for x, y ∈ R≥0 , we have √xy ≤ (x + y)/2.(Hint: (x − y)2 ≥ 0 by Exercise 4.46. Use algebrai manipulation to make this inequality look like the...

Prove the Arithmetic Mean–Geometric Mean inequality: for x, y ∈ R≥0 , we have √xy ≤ (x + y)/2.(Hint: (x − y)2 ≥ 0 by Exercise 4.46. Use algebrai manipulation to make this inequality look like the desired one.)

3.Prove that the arithmetic mean and geometric mean of x and y are equal if and only if x = y.

Exercise 4.46

Let x ≥ 0 and y ≥ 0 be arbitrary real numbers. The arithmetic mean of x and y is (x + y)/2, their average. The geometric mean of x and y is √xy

1.First, a warm-up exercise: prove that x2 ≥ 0 for any real number x. (Hint: yes, it’s easy.)