I am performing a logistic regression with some continuous and categorical variables. I introduced a variable and its squared so that I can compute a threshold from which the sign of the marginal effect changes. I know how to do this with OLS regressions. For example, if I want to see if the sign of the impact of $TEST$ variable changes : $$ y_i = \beta TEST_i + \gamma TEST_i^2 $$ then $$ \frac{\delta y_i}{\delta TEST_i} \gt 0 $$ if $$ TEST \gt \frac{- \beta}{2 \gamma} $$ That gives the threshold from which the sign changes. However, for logistic regression, we are estimating (with $TEST$ a continuous variable) : $$ Prob(y_i = 1) = \frac{e^{(\beta TEST_i + \gamma TEST_i^2)}} {1 + e^{(\beta TEST_i + \gamma TEST_i^2)}} $$ From here, how can I compute the value from which the impact of the variable changes ?

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