Mathematics

Problem

Let

f(x)=\ln x

for

x>0

and consider the interval

[1,e]

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

f

[1,e]

. b. (ii) Determine all real numbers

c

(1,e)

such that

f'(c)

equals the average rate obtained in part (i). c. (iii) Prove that exactly one such number

c

exists in

(1,e)