A globally convergent fixed point method can be defined for functions satisfying second order ODEs y_n^''+B(x)y_n^'+A_ny_n=0 (being n the order) and verifying linear formulas, with continuous coefficients, relating its derivative with the same function and a function of a contiguous order. Theses conditions are met by a broad family of special functions (solutions of Bessel, Coulomb, Hermite, Legendre and Laguerre equations, among others) and leads to efficient algorithms for the computation of the zeros of general solutions of these equations. We show that similar techniques can be applied for the computation of turning points. The interlacing between the zeros of a solution of a second order ODE y_n^''+A_ny_n=0 and its derivatives when A_n>0 (at most, there can be only a zero and there can be two turning points when A_n0) and the monotonicity of the logarithmic derivative y_n^'/y_n is the starting point in this case. Computer algebra systems help in automatically performing the transformations required prior to the application of the fixed point methods.