Does $f(x)=ax$ intersect $g(x)=\sqrt{x}$ It maybe a stupid question but I want to be sure how to explain it formally.
Does $f(x)=ax$ intersect $g(x)=\sqrt{x}$, when $x>0$ and $a>0$ (however small it is)
I think it does. 
The derivative of $f(x)$ is constant, positive. And the derivative of $g(x)$ tends to $0$. So there will be some point $x_0$, from which the derivative of $f$ will be greater than derivative of $g$. Therefore $g$ will grow slower than $f$ and both functions finally meet. Am I right? This is enough? Can one formally prove it? 

writing $f(x) = g(x)$ you arrive at the equation $a^2 x^2 = x$, which gives you two solutions $x=0$ and $x=1/a^2$.
