Question 162
a) Show that for all $x \gt 1$ the following inequality holds: $2^{x} \cdot \sqrt{4^{x}-1}+4^{x} \cdot \sqrt{2^{x}-1}\lt 8^{x}$.
b) Determine the bijective functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x, y \in \mathbb{R}$ the relation $f(2 x+f(x)+3 f(y))=f(3 x)+f(3 y)$ holds.
b) Determine the bijective functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x, y \in \mathbb{R}$ the relation $f(2 x+f(x)+3 f(y))=f(3 x)+f(3 y)$ holds.