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

The question reads:

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.

Any tips are appreciated.

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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 $$(\sqrt{3}+\sqrt{5})^2 = 8+2\sqrt{15} > 8+2\sqrt{12} = (\sqrt{2}+\sqrt{6})^2.$$

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  • $\begingroup$ I have to agree that proving such inequalities using MVT is a bit lame application of MVT. $\endgroup$ – Paramanand Singh Dec 12 '13 at 5:37

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