DE equilibrium points -- am I on the right track?

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: We're given the eq. dy/dt = f(y) : suppose that y1 is a critical point: f(y1) = 0 : We're asked to show that the constant equilibrium solution g(t) = y1 is: : asymptotically stable if f'(y1) < 0 : asymptotically unstable if f'(y1) > 0 : ------------------------------ : I think I know how to solve this but need a sanity check... : f'(y1) is the derivative of f(t). : if f'(y1) is negative, then it means the slope of f(t) is downward around the point y1, and since y1 is a zero of f(y) then f(y) must be positive when y < y1 and negative when y > y1. Thus it is asymptotically stable at y1. : if f'(y1) is > 0, then it's just the opposite and asymp. unstable. : is this a satisfactory & complete answer? : or is there some more rigorous formal way to prove this?

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