Hermite/Lagrange Polynomials

Suppose you have the following data for a function

xi = [0.3, 0.4, 0.5, 0.6]

yi = exp.(-xi) .- xi

= [0.440818220681717, 0.27032004603563, 0.10653065971263, -0.051188363905973 ]

That is, yi is the function e-x - x evaluated at the four values in the vector xi.

You can find the root of the equation e-x = x by making a function interpolating through the points (yi, xi) and evaluating that function at y = 0. When I did this calculation using a Lagrange interpolating polynomial, I estimated the root to be x = 0.5671423536. The exact value of the root to 10 decimal places is 0.5671432904.

(a) Is this a good way to find a root in general? If so, why; if not, why not?

(b) Would it be helpful to use a Hermite interpolating polynomial instead of Lagrange?

(c) What method, combined with these results, would be a good way to improve this estimate of the root?