In toxicology studies, dose response functions with a downturn at higher doses are often observed. For such response functions, researchers often want to see if the downturn of the response is signifcant. A probit model with a quadratic term is adopted to demonstrate the dose response with a downturn. Under the probit model, we obtain optimal designs to study the signifcance of the downturn and their efficiencies are compared. Our approach identites the upper bound of the number of optimal design points and searches for the optimal design numerically based on the upper bound.