Problem 28P

28. (a) The cubic function $f(x)=x(x-2)(x-6)$ has three distinct zeros: 0,2, and $6 .$ Graph $f$ and its tangent lines at the average of each pair of zeros. What do you notice?

(b) Suppose the cubic function $f(x)=(x-a)(x-b)(x-c)$ has three distinct zeros:

$a, b$, and $c .$ Prove, with the help of a computer algebra system, that a tangent line drawn at the average of the zeros $a$ and $b$ intersects the graph of $f$ at the third zero.

Step-by-Step Solution