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
