   Chapter 2.P, Problem 28P

Chapter
Section
Textbook Problem

# (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.

To determine

(a)

To sketch: The graph of f and its tangent lines at the average of each pairs of zeros and to explain what will notice

Explanation

1) Concept:

Take the average of each pair of zeros and then draw the tangent line at that point and then conclude.

2) Given:

fx=x(x-2)(x-6)

3) Graph:

Consider the purple tangent line at

x=2+62=4,

X intercept of this line is at x = 0 which is third zero of the cubic function

Now, consider blue tangent at

<

To determine

(b)

To prove:

The tangent line drawn at the average of zeros a and b intersects the graph of f at the third zero

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