<s>[INST] Finding the maximum value without using derivatives 
Find the maximum value of $$f(x)=2\sqrt{x}-\sqrt{x+1}-\sqrt{x-1}$$ without using derivatives.

The domain of $f(x)$ is $x \in [1,\infty)$. Then, using derivatives, I can prove that the function decreases for all $x$ from $D(f)$ and the maximum value is $f(1)= 2 - \sqrt{2}$. However, this uses derivatives.
 [/INST] Note that
\begin{align}
f(x) & = (\sqrt{x}-\sqrt{x-1})-(\sqrt{x+1}-\sqrt{x})\\
& = \frac{1}{\sqrt{x}+\sqrt{x-1}} - \frac{1}{\sqrt{x+1}+\sqrt{x}}\\
& = \frac{\sqrt{x+1}-\sqrt{x-1}}{(\sqrt{x}+\sqrt{x-1})(\sqrt{x+1}+\sqrt{x})} \\
& = \frac{2}{(\sqrt{x}+\sqrt{x-1})(\sqrt{x+1}+\sqrt{x})(\sqrt{x+1}+\sqrt{x-1})} 
\end{align}
which is a decreasing function of $x$.
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