we know that we use the following formula to find a turning point in income inequality researches:
Gini=B0 + B1gdp + B2gdp^2 + e
u = exp(-B1/2B2)
Gini=B0 + B1gdp + B2gdp^2 + B3gdp^3 + e;
For the cubic equation there will be two turning points as the derivative will be a quadratic equation.
this is the answer if you look for an answer for that equation
This sort of polynominal might have several turning points, depending on the value of the parameters.
https://undergroundmathematics.org/calculus-meets-functions/r7861/solution
https://sciencing.com/calculate-slope-regression-line-8139031.html
I would first plot it for the relevant values of GDP in your population and then compute the turning or turning points you are interested in.
I hope it helps,
José-Ignacio
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