In Stone and Goldbart's "Mathematics for Physics" and they said:
Suppose that we have a differentiable function $J(y_1,y_2, . . . ,y_n)$ of $n$ variables and seek its stationary points — these being the locations at which $J$ has its maxima, minima and saddlepoints. At a stationary point $(y_1,y_2,...,y_n)$ the variation: $$\delta J= \sum_{i=1}^n\frac{\partial J}{\partial y_i}\delta y_i $$ must be zero for all possible $\delta y$.
Why does it have to be zero? What are these stationary points and why are they important? Thank you.
This is essentially a result from calculus. If a function $f(x)$ has a maximum or a minimum (or a saddle) point, then its derivative is zero. If the derivative in some coordinate direction was not zero then it the function would either be increasing or decreasing along that direction, which is impossible for maxima, minima, or saddle points. Thus for all $i$,
$$\frac{\partial J}{\partial y_{i}} = 0$$