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)).

**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 stable equilibrium.

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