Gradient Descent on differential equation

Question

I have a differential equation of the form $$\frac{d}{dx}f=f^2$$ I want to find a root of the second derivative of $f$, in order to maximize the derivative $df/dx$.

I could of course simply solve the differential equation and apply gradient descent on the second derivative to find the root, but I would like to find this root without having to solve the differential. Any thoughts?

Answer 1

$\frac{df}{dx}=f^2 \Rightarrow \frac{d^2f}{dx^2}=2f\frac{df}{dx}=2f^3$

$f(x)=0 \Rightarrow \frac{d^2f}{dx^2}=0$

Answer 2

You have

$$f^\prime(x)= f^2(x)$$ hence taking the derivative on both sides

$$f^{\prime \prime}(x)=2f(x)f^\prime(x)=2f^3(x)$$which proves that the second derivative vanishes if and only $f$ vanishes at the same point.

But if $f$ vanishes at $x_0$, you also have $f^\prime(x_0)=0$. And according to the Picard-Lindelöf theorem, $f$ is the always vanishing map.

Conclusion: either $f$ is the always vanishing map or its second derivative never vanishes as $f$ itself.