Consider the polynomial function .

MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation

Problem

Consider the polynomial function

g(x)=x^{3}-6x^{2}+9x+2

. (a) Compute the average rate of change of

g

on the interval

[1,4]

. (b) Show that there exists at least one value

c

(1,4)

such that

g'(c)

equals the average rate found in part (a), and determine all such values of

c

. Provide algebraic justification. (c) Write the equation of the tangent line to the graph of

g

at each value of

c

obtained in part (b). (d) Determine

g''(c)

for each

c

found above and explain how the sign of

g''(c)

clarifies the relationship between the tangent line at

x=c

and the secant line through

(1,g(1))

and

(4,g(4))

. Connect your explanation explicitly to concavity.