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Solve $3^x + 6^x = 5^x + 4^x$ using Lagrange's mean value theorem's.

I believe all I have to do is apply Lagrange's theorem on some function on a given interval, but I know neither the function to apply on nor the interval.

So, please, help me if you know how.

Thank you in advance!

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A big start to the answer.

You are trying to solve: $6^x-5^x=4^x-3^x$. The mean value theorem says that $6^x-5^x=xa^{x-1}$ for some $a\in [5,6]$, and that $4^x-3^x=xb^{x-1}$ for some $b\in [3,4]$.

So, given that you known $a\in [5,6]$ and $b\in [3,4]$, when is it possible that:

$$xa^{x-1}=xb^{x-1}?$$

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