Mathematics

Problem

Consider the function

f(x)=x^{3}-3x+2

on the closed interval

[1,2]

. a. (i) Calculate the average rate of change of

f

on the interval

[1,2]

. b. (ii) Obtain

f'(x)

from first principles or otherwise. c. (iii) Find all numbers

c

in the open interval

(1,2)

for which

f'(c)

equals the average rate found in part (i). Show all necessary algebraic steps.