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

Disucssion:

$$ U(x)=k(2x^3-5x^2+4x)$$
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 stble equilibrium.