Hint:Using the supplied data, derive a particle acceleration relationship. In this equation, substitute the value in the form of a change in velocity. Substitute the value of time change in this equation as well. Integrate this equation between the velocity and displacement limits to get the relationship between the change in kinetic energy of the particle and its displacement.Step 1:Find the loss of kinetic energy for any displacement \\\\(x\\\\) is proportional to.Let,Net work done by all the forces give the change in kinetic energy\\\\(\\Rightarrow\\\\)\\\\(W=\\frac{1}{2} m v^{2}-\\frac{1}{2} m v_{0}^{2}\\\\) \\\\(W=k_{f}-k_{i}\\\\)where,\\\\(m=\ ext { mass of the body }\\\\)\\\\(v_{0}=\ ext { initial velocity }\\\\)\\\\(v=\ ext { final velocity }\\\\)Step 2:Retardation \\\\(a \\alpha-x\\\\)\\\\(\\Rightarrow\\\\)\\\\( \\frac{d v}{d t}=-k x \\quad[\\mathrm{k}\\\\) is proportionality constant\\\\(\\Rightarrow\\\\)\\\\( v \\frac{d v}{d x}=-k x\\\\)\\\\(\\Rightarrow\\\\)\\\\(\\int_{v_{1}}^{v_{2}} v d v=-k \\int_{0}^{x} x d x\\\\)\\\\(\\Rightarrow \\frac{1}{2}\\left(v_{2}^{2}-v_{1}^{2}\\right)=-\\frac{1}{2} k x^{2}\\\\)Loss in kinetic energy \\\\(=\\frac{1}{2} m\\left(v_{2}^{2}-v_{1}^{2}\\right)=-\\frac{1}{2} k x^{2}\\\\)Loss in kinetic energy \\\\(\\alpha x^{2}\\\\)Therefore, The correct option is 'C' is \\\\(x^{2}\\\\).