For \(U(x)=4x\), find \(F_x\).

For \(U(x)=\frac{1}{2}kx^2\), find \(F_x\).

If \(U(x)\) increases as \(x\) increases, which direction does the force point?

What condition identifies an equilibrium point on a one-dimensional potential graph?

For \(U(x)=3x^2\), find \(F_x\) at \(x=2\,\mathrm{m}\).

For \(U(x)=x^3-6x\), find \(F_x(x)\).

For \(U(x)=2x^2-8x\), find the equilibrium point.

For \(U(x)=2x^2-8x\), classify the equilibrium at \(x=2\,\mathrm{m}\).

For \(U(x)=x^4-4x^2\), find all equilibrium positions.

Classify the equilibrium positions for \(U(x)=x^4-4x^2\).

For \(U(x)=ax^4-bx^2\), where \(a>0\) and \(b>0\), find \(F_x(x)\), find and classify all equilibrium points, and determine the barrier height between the stable equilibria.

A particle of mass \(m\) has \(U(x)=\frac{A}{x^2}+Bx\) for \(x>0\), with \(A>0\) and \(B\) allowed to be positive, zero, or negative. Derive the condition for a stable equilibrium to exist, find its position if it exists, and derive the small-oscillation angular frequency about it.