# Mean Value Theorem to estimate values of square roots

I have a question related to the Mean Value theorem and it's application, and I'm really stumped.

Use the Mean Value Theorem and the property of the function $f(x) =\sqrt{x}$ and its derivatives $f'(x)$ and $f''(x)$ to find out which is larger: √3+√5 or √2+√6 without actually calculating their numerical values.
I think they want you to use the mean value theorem to compare $\sqrt{3}-\sqrt{2}$ to $\sqrt{6}-\sqrt{5}$, by taking advantage of the fact that $f'$ is monotonic decreasing.
But that is far from the most direct way to solve the problem: instead, just notice that $$(\sqrt{3}+\sqrt{5})^2 = 8+2\sqrt{15} > 8+2\sqrt{12} = (\sqrt{2}+\sqrt{6})^2.$$