How to find if equilibrium points of a force is un/stable?

[Blockade]

Homework Statement

U = Ax² - Bx³

Homework Equations

du/dx = 2Ax - 3Bx²

The Attempt at a Solution

If I was given a potential energy function U = Ax² - Bx³ and am asked to find:

1) The expression for the force as a function of x.

2) The equilibrium points and determine if are they stable or unstable?

So, for 1):
Would I differential the function giving like so?

U' = f'(x) = 2Ax - 3Bx²

Now for 2):
Would I set f'(x) = 0 to find the equilibrium points?

f'(x) = 2Ax - 3Bx² = 0

In return I get the points of x through the quadratic equation:
x = 0 and X = A/B

If this is all correct how can I determine if a equilibrium point is stable or unstable?

 

[ehild]

Homework Statement

U = Ax² - Bx³

Homework Equations

du/dx = 2Ax - 3Bx²

The Attempt at a Solution

If I was given a potential energy function U = Ax² - Bx³ and am asked to find:

1) The expression for the force as a function of x.

2) The equilibrium points and determine if are they stable or unstable?

So, for 1):
Would I differential the function giving like so?

U' = f'(x) = 2Ax - 3Bx²

Recall the force is negative derivative of the potential energy.

Now for 2):
Would I set f'(x) = 0 to find the equilibrium points?

f'(x) = 2Ax - 3Bx² = 0

In return I get the points of x through the quadratic equation:
x = 0 and X = A/B

If this is all correct how can I determine if a equilibrium point is stable or unstable?

X=0 is correct, but you have a mistake in the other equilibrium point.

The equilibrium is stable if the potential energy is minimum in that point and unstable if it is maximum.

 

[Blockade]

Recall the force is negative derivative of the potential energy.

X=0 is correct, but you have a mistake in the other equilibrium point.

The equilibrium is stable if the potential energy is minimum in that point and unstable if it is maximum.

Oh sorry, x = 2A/(3B).
How I find the max and min of potential energy?

 

[ehild]

Oh sorry, x = 2A/(3B).
How I find the max and min of potential energy?

You found the positions of the extremes, at x=0 and at x=2A/3B.
Have you learned what should be the second derivative at a maximum and at a minimum?