A potential energy function is given by u (x) = (3.00n) xâ (1.00n/m2) x3. at what position or positions is the force equal to zero?

I believe the correct form of the energy function is:

u (x) = (3.00 N) x + (1.00 N / m^2) x^3

or in simpler terms without the units:

u (x) = 3 x + x^3

Since the highest degree is power of 3, therefore there are two roots or solutions of the equation.

Since we are to find for the positions x in which the force equal to zero, u (x) = 0, therefore:

3 x + x^3 = u (x)

3 x + x^3 = 0

Taking out x:

x (3 + x^2) = 0

So one of the factors is x = 0.

Finding for the other two factors, we divide the two sides by x and giving us:

x^2 + 3 = 0

x^2 = - 3

x = sqrt ( - 3)

x = - 1.732 i, 1.732 i

The other two roots are imaginary therefore the force is only equal to zero when the position is also zero.

Answer:

x = 0