Problem: Finding the maximum speed of a 2kg particle free to move along x-axis, given its potential energy and total mechanical energy.

Given:
- Potential Energy, V(x) = {(x^3)/3 - (x^2)/2}J
- Total Mechanical Energy, E = 4J
- Mass of the particle, m = 2kg

Solution:
To find the maximum speed of the particle, we need to use the conservation of energy principle, which states that the total mechanical energy of a system is conserved if there are no external forces acting on it.

Step 1: Finding the Kinetic Energy Function
The kinetic energy function can be found by using the conservation of energy principle.
- We know that the total mechanical energy of the system is given by E = KE + PE, where KE is the kinetic energy and PE is the potential energy.
- The potential energy function is given to us as V(x) = {(x^3)/3 - (x^2)/2}J.
- Therefore, the kinetic energy function can be found by KE = E - PE.
- Substituting the given values, we get KE = 4 - {(x^3)/3 - (x^2)/2}J.

Step 2: Finding the Velocity Function
The velocity function can be found by using the kinetic energy function.
- We know that KE = (1/2)mv^2, where v is the velocity of the particle.
- Substituting the value of KE, we get 4 - {(x^3)/3 - (x^2)/2} = (1/2)(2)v^2.
- Simplifying the equation, we get v^2 = 2(4 - {(x^3)/3 - (x^2)/2}).
- Therefore, the velocity function is given by v(x) = sqrt(8 - 2(x^3)/3 + x^2).

Step 3: Finding the Maximum Velocity
The maximum velocity of the particle can be found by finding the maximum value of the velocity function.
- To find the maximum value, we need to take the derivative of the velocity function and set it to zero.
- Differentiating the velocity function with respect to x, we get dv/dx = (-2x^2 + 2x)/sqrt(8 - 2(x^3)/3 + x^2).
- Setting dv/dx to zero, we get -2x^2 + 2x = 0.
- Solving for x, we get x = 0, x = 1.
- To find which value of x gives the maximum velocity, we need to take the second derivative of the velocity function and evaluate it at x = 0 and x = 1.
- Differentiating dv/dx with respect to x, we get d^2v/dx^2 = -2(2x^3 - 3x^2)/((8 - 2(x^3)/3 + x^2)^(3/2)).
- Evaluating d^2v/dx^2 at x = 0 and x = 1, we get d^2v/dx^2 at x = 0 is negative and d^