Using maxima / minima to find roots 
Suppose $c$ is a given positive number. The equation $\ln x=cx^2$ must
  have a solution if
A) $c<\frac{1}{2e}$
B) $c>\frac{1}{2e}$
C) $c<\frac{1}{e}$
D) $c>\frac{1}{e}$

I have no idea how to approach this, my professor used a method that used the maximum of $\ln x - cx^2$, but I could not understand it.
Any help will be appreciated.

HINT: defining $$f(x)=\ln(x)-cx^2$$ and we get $$f'(x)=\frac{1}{x}-2cx$$
can you proceed?
