Number of real roots of $3^x+4^x=2^x+5^x$ with proof This equation 
$$3^x+4^x=2^x+5^x$$
has two obvious real roots. The question is if it has more real roots than two. A proof is required in any case. 

We consider the equation
$$3^x+4^x=2^x+5^x~~~~(1)$$
Use Lagranges Mean Value Theorem (LMVT) for the function $f(t)=t^x$ for two intervals $(2,3)$ and $(4,5)$. So 
$$\frac{3^x-2^x}{3-2}=xt_1^{x-1}, ~~~t_1 \in (2,3)~~~~(2)$$
and
$$\frac{5^x-4^x}{5-4}=xt_2^{x-1}, ~~~t_2 \in (4,5)~~~~(3).$$
By equating (2) and (3), we get (1) and 
$$xt_1^{x-1} = xt_2^{x-1}, ~~~t_1 \ne t_2\Rightarrow x=0~ \mbox{or}~ x=1.$$
Hence Eq. (1) can have only two real roots.
