91Ó°ÊÓ

If \(u>0\) is any real number and \(x

Prove that \(\left[\frac{1}{2}(a+b)\right]^{2} \leq \frac{1}{2}\left(a^{2}+b^{2}\right)\) for all \(a, b \in \mathbb{R}\). Show that equality holds if and only if \(a=b\)

(a) Prove there is no \(n \in \mathbb{N}\) such that \(0

(a) If \(c>1\), show that \(c^{n} \geq c\) for all \(n \in \mathbb{N}\), and that \(c^{n}>c\) for \(n>1\). (b) If \(01\).

If \(a>0, b>0\), and \(n \in \mathbb{N}\), show that \(a