Laplace method when all derivatives at the critical point are 0

Question

When trying to estimate by the Laplace method the simple integral $ \int_0^1 e^{-n f(x)}dx $ with $ f(x) := \frac{1}{x} + \frac{1}{1 - x} - 4 $, I ended up realising that all the derivatives at the extrema $ x^* = 1/2 $ are in fact equal to $0$. The true case I am considering being more complicated (in higher dimension), I was wondering : what happens if all the higher order derivatives taken at the critical point are $0$ ? Is there a standard reference for such a problem ?