Let x* be the solution of the equation h(x*) = 0. Show that the critical point of a generalized first-order polynomial is x* - 1 and the point of inflection is x* - 2.
Polynomials form a useful set of functions in part because the derivative of a polynomial is another polynomial. Another set of functions with this useful property is called the generalized polynomials, formed as products of polynomials and exponential functions. One simple group of generalized polynomials are the products of linear functions with the exponential function, taking the form
h(x) = (ax + b)ex
for various values of a and b. We will call these generalized first-order polynomials.