Optimized Path of a Particle experiencing a Time - Varying Force

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Homework Statement

Given a force F(t), calculate the optimized path the particle, initially at a, should take to reach point b. Consider the particle to travel in a one - dimensional space only (x axis). The particle has kinetic energy only.

Homework Equations

L = T - U
$\frac{\partial L}{\partial x_{i}} - \frac{d}{dt} \frac{\partial L}{\dot{x}_{i}} = 0$

The Attempt at a Solution

If the particle is only purely kinetic, then the first term of the equation is zero and the second term is only $\frac{d}{dt} mv(t)$ which implies that the velocity is constant in time. I also do not know if this will give the optimized path of the particle and how to implement it...