Invariance and instability

In the paper, the problem of stability of the zero solution of an autonomous system of differential equations is considered. The problem of maximal improvement of a function with the derivative of constant sign is solved which the additional functions method can permit. The theorem on the construction of a function with definite derivative or with the derivative of constant sign is obtained for which the set of derivative's zeros is invariant. The theorem of instability is proved for the case of the existance of a derivative of constant sign. Illustrative examples are considered.