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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.