How many solutions of the equation $[f'(x)]^2=f(x)\cdot f''(x)$.

Question

A function of degree 4 has the graph as shown. How many solutions of the equation $[f'(x)]^2=f(x)\cdot f''(x)$. The answer is $0$ solution. I have tried to express as: $$\int\frac{f'(x)}{f(x)}dx=\int\frac{f''(x)}{f(x)}dx$$ Which gives $\ln|\frac{f(x)}{f'(x)}|=C$. But I found no relation to the answer. Hope your kind help!

Answer 1

You are looking for points where $$ \frac{d^2}{dx^2}\ln|f(x)|=\frac{d}{dx}\frac{f'(x)}{f(x)}=\frac{f(x)f''(x)-f'(x)^2}{f'(x)^2}=0 $$ But for a polynomial of degree $4$ with $4$ distinct roots, $f(x)=a\prod_{k=1}^4(x-x_k)$, you get $$ \frac{d^2}{dx^2}\ln|f(x)|=\sum_{k=1}^4\frac{1}{(x-x_k)^2} $$

which is always positive or infinite as sum of squares.