Then you should know to evaluate your original function at the roots of its derivative having odd multiplicity.
I have computed approximations for the 3 roots of the cubic polynomial using Newton's method to 12 digits, given you a link to an article about how the method works, given you a link to an article about how to obtain the exact value of the roots. I'm happy with the approximations...I'm not going to use the cubic formula...it is messy and time consuming, and not worth the effort, in my opinion. Substituting the huge cumbersome values into the original function would not be fun either.
So, take the roots of the derivative of odd multiplicity, which have all been found, and evaluate the original function at these roots. These are your turning points.