Using maxima / minima to find roots

Question

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.

Answer 1

Define $f(x)=$In$(x)-cx^2$, $c>0$ on $x\in (0,\infty)$.

$f'(x)=0$ gives you $x=\frac{1}{\sqrt{2c}}$ as the stationary point in $(0,\infty)$. Since $f''(\frac{1}{\sqrt{2c}})<0$, so $x=\frac{1}{\sqrt{2c}}$ is point of maxima.

As $f(0)<0$ so you just need that $f_{max}>0$ in order to make the graph of $f$ to cross the X-axis.

So $f_{max}=f(\frac{1}{\sqrt{2c}})>0$ will give you $c>\frac{1}{2e}$

Answer 2

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