you may add a constraint on those parameters, thus the derivative dy/dx is positive all the time, namely, you would solve for those b's with an extra condition that (b1+2*b2*x+3*b3*x^2+4*b4*x^3) >=0 for all x.
you may still end up with many solutions, so you may need other additional conditions if you're asking for a unique solution.

Many thanks for your kindness. Could you please offer a code so that I can copy and run it? I am a green hand, you know.
Best regards

Sorry i don't program in Matlab. btw, i don't think exp() is a good fit to your data. y=log(polynomial(x)) could be a candidate since it saturates when x is large.

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