In the Feynman lectures on physics, Feynman in talking about the principle of least action, discusses how we should be able to find the true path $x(t)$ which has the least action, and the way to do it he says:
When we have a quantity which has a minimum—for instance, in an ordinary function like the temperature—one of the properties of the minimum is that if we go away from the minimum in the first order, the deviation of the function from its minimum value is only second order. At any place else on the curve, if we move a small distance the value of the function changes also in the first order. But at a minimum, a tiny motion away makes, in the first approximation, no difference (Fig. 19–8).
“That is what we are going to use to calculate the true path. If we have the true path, a curve which differs only a little bit from it will, in the first approximation, make no difference in the action. Any difference will be in the second approximation, if we really have a minimum.
The thing I understand from the diagram is that, near the minimum region the curve can be approximated as a parabola. Anywhere else on the curve, a tiny segment can be modeled as a straight line.
I do not understand what he means by first and second approximations, and that in the first approximation there is no difference, but there is a difference in the second approximation.