2 Suppose that g is continuous on ab and that g has three ze

Solution

If g\" is continuous, it must vary smoothly in the interval. For there to be three zeros, one of two cases must hold in the interval. Either the slope changes from plus to minus to plus or the slope changes from minus to plus to minus. Two of the zeros could also be points of inflection at the end points of the interval, but this would mean that there were points of inflection at the end points and that the third must be in the open interval by the \"no choice theorem.\" The three possibilities being 1) the beginning point of the closed interval, 2) the enclosed open interval in between, and 3) the ending point of the closed interval. This could be called the \"ruler postulate.\" The third zero must be greater than the smallest point and less than the greatest point. Generalization: If there are n zeros in the closed interval, g\" must have at least n-2 zeros in the open interval.