i have searched and found this formulation for cubic fit
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Use calculus. To find min(f(x)) then solve . The roots of that give you the critical points; call those for the moment. Then, examine . Positive values indicate that the corresponding was a minima.
For a degree 4 polynomial, the derivative is degree 3, and there are closed form solutions for the roots of a polynomial of degree 3. Therefore you can calculate the exact critical points, and test them.
You will only find one real-valued root. You might assume that it must be the minima, but in the general case, it will not be a safe assumption.
Are you supplying your own derivative calculations using the GradObj option (and Hessian option if applicable)? You should do so, since the analytical derivatives are easy here. With lsqcurvefit, there are similar options, e.g. Jacobian.
How are you initializing the optimization? Because your model is loglinear, it is likely that the initial guess as generated below will be more effective than random guessing.
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