Difficult exponential equation $6^x-3^x-2^x-1=0$ $$6^x-3^x-2^x-1=0$$I don't have any idea how to solve this equation. I don't know how to show that the function $f(x)=6^x-3^x-2^x-1$ is strictly increasing (if it's the case). The derivative doesn't show me anything. I tried also to use the Lagrange theorem(like in this equation for example: $2^x+9^x=5^x+6^x$), but it's not the case!
But I need a solution without calculus. Please help!
Edit: In my country the theorem that states roughly this $f'(c)=\dfrac{f(b)-f(a)}{b-a}, $where $f:[a,b]\to\mathbb{R} $ is called Lagrange theorem

$6^x-3^x-2^x-1=0$ equivalent $1=\frac 1 {2^x}+\frac 1 {3^x}+\frac 1 {6^x}$
The left side is constant while the right side is decreasing. There can only be a solution at most, and $x=1$ is perfect for the job.
