Question 69
Let $\mathbb{R}^{+}$ be the set of all positive real numbers and $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ be a function such that
$$
f(x+y)-f(x-y)=4 \sqrt{f(x) f(y)}
$$
for all $x\gt y\gt 0$.
a) Prove that $f(2 x)=4 f(x)$ for all $x \in \mathbb{R}^{+}$.
b) Find all such functions.
$$
f(x+y)-f(x-y)=4 \sqrt{f(x) f(y)}
$$
for all $x\gt y\gt 0$.
a) Prove that $f(2 x)=4 f(x)$ for all $x \in \mathbb{R}^{+}$.
b) Find all such functions.