Three forces (A, B. and C) act on a mass (the black dot) was the mass undergoes

Question

Three forces (A, B. and C) act on a mass (the black dot) was the mass undergoes a displacement Delta r to the right. Rank the values of the work done by these three forces from smallest value (most negative) to largest (most positive). A force F = 12i - 9j N acts on a particle in the xy plane. The particle undergoes a displacement of Delta r = 2i -4j m. Please calculate the work done by this force over the displacement given. The potential energy of a particle on the x-axis is given by U = 6 + 8xe^-3x^2, where .r is in meters and U is in Joules. Please find the first point on the positive side of the x axis for which the potential is a maximum or minimum (that is, find the equilibrium position for x > 0). Is the point that you found in the question immediately above a point of stable equilibrium or unstable?

Explanation / Answer

(1)
As the displacement happens towards right. Therfore work done by C is Highest , While Work done by B is Smallest.
Therefore -
B<A<C

(2)
Work done = Force . Displacement
Work done = (12i^ - 9j^) .(2i^ - 4j^)
Work done = 24 + 36
Work done = 60 J

(3)
U = 6 + 8x e^-3(x^2)
dU/dx = 0 + 8 *e^-3(x^2) + 8x (-6x) *  e^-3(x^2) = 0
8 *e^-3(x^2) - 8*6 x^2 *  e^-3(x^2) = 0
1 - 6x^2 = 0
x =  0.41 m

4.) Unstable

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