Chapter 4.1, Problem 57ES

(a) Prove that a general cubic polynomial $f\left(x\right)=a{x}^3+b{x}^2+cx+d\left(a\ne 0\right)$

(b) Prove that if a cubic polynomial has three x-intercepts, then the inflection point occurs at the average value of the intercepts.

(c) Use the result in part (b) to find the inflection point of the cubic polynomial $f\left(x\right)=x^3-3x^2+2x,$ and check your result by using $f^{″}$ to determine where $f$ is concave up and concave down.