The second derivative is zero tells us nothing about stability.
What is a solution's equilibrium?
Differential equations can have solutions called equilibrium solutions when the derivative along those solutions equals zero. In other words, at that solution, the slope is a horizontal line.
Consider the following potentials:
U(x) = [tex]x^{4}[/tex]
U(x ) = [tex]x^{6 } - x^{4}[/tex]
U( x ) = [tex]x^{4} + x^{3}[/tex]
All three of these potentials have an equilibrium point at x=0. All three of these potentials are such that the second derivative of U(x) at this equilibrium point is zero.
However, you should convince yourself (perhaps by plotting these potentials) that in the first case the equilibrium is stable, in the second case it is unstable, and in the third case the equilibrium is, as you put it, "stable in one direction but unstable in the other".
The second derivative is zero tells us nothing about stability.
Learn more about equilibrium
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