# Solve exponential equation using the mean value theorem

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.

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