let be 3 positive roots of the equation where .
show that there is a root of the equation in
It depends on what you can use (any methods of calculus?) and the level of strictness. Basically, if you look at the graph of , this is obvious, but the question is, how much handwaving you are allowed to use in referring to the graph.
The function behaves like in that it is positive on and negative on for and it is zero in , . However, the amplitude increases exponentially. You can see the graph in WolframAlpha. So if are three positive roots of , then there is an interval either inside or inside (or both, if these are not consecutive roots), where is negative. Also, one can show that even in the first negative dip between and , the function's minimum is far less than -1/2.