Characterizations of Quadratic Polynomials

SummaryThis note was motivated by a student's misconception that the average velocity over a time interval of a particle in rectilinear motion is the arithmetic mean of its velocities at the ends of the interval. We present an algebraic proof that this property holds only if the particle has constant acceleration, thereby recovering a well-known result. A characterization is also provided of differentiable functions on (-∞, ∞) whose restrictions to the intervals (-∞, 0] and [0, ∞) are quadratic polynomials. In addition, a calculus-free proof is presented to show that a certain feature of the mean value theorem holds only for quadratic polynomials.