Let’s discuss a problem where we find a stable equilibrium point, useful for Physics Olympiad.

The Problem:

Find the stable equilibrium point corressponding to the potential (U(x)=k(2x^3-5x^2+4x)).


$$ U(x)=k(2x^3-5x^2+4x)$$
Differentiating with respect to x
$$ \frac{dU}{dx}=k(6x^2-10x+4)
Differentiating again,
$$ \frac{d^2U}{dx^2}=k(12x-10)
Putting (x=1) $$ \frac{d^2U}{dx^2}=+2k$$
This is positive because (k) is positive and it is the minimum hence the (x=1)
corresponds to stable equilibrium.