What is a second-type-touch for curve?

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Let I have some f: R^2 -> R. And M is a zero-set point of f. Suppose f is infinity differenced and gradient is not 0 forever. If f(x0, y0)=0 how can I define second-type-touch from f_x(x0, y0), f_y(x0, y0), f_xx(x0, y0), f_xy(x0, y0) and f_yy(x0, y0)?