The potential energy of an object as a function of position is U(x)= x² - x - 6,where U is measured in joules and x is measured in meters.Where is the object be located if it is in a stable equilibrium?

Question

Quizplus · Accepted Answer

To find the point of stable equilibrium, we need to look for the minimum of the potential energy function. 
Taking the derivative of the given function with respect to x, we get: 
U'(x) = S1U1P12S1S1P0 - 1 
Setting U'(x) = 0 to find the critical points, we get: 
S1U1P12S1S1P0 - 1 = 0 
Simplifying, we get: 
x = (S1U1P12S1S1P0 - 1)/S1S1P0 
Since U''(x) = -1 (the second derivative is negative), we can confirm that this critical point is a minimum, which is the location of the stable equilibrium. 
Plugging in the value of U1, we get: 
x = (S1S1P0 - 1)/S1S1P0 
x = (1/2)/1 
x = 1/2 
Therefore, the object is located at x = 0.5 m in a stable equilibrium.

The Potential Energy of an Object as a Function of Position