Mathematics

Problem

Consider the functions

f(x)=\ln(1+x)

and

g(x)=\sqrt{x}

defined for

x\ge 0

. a. (i) Verify that

f(0)=g(0)

. b. (ii) Find the set of all

x\ge 0

for which

f'(x)>g'(x)

. c. (iii) Using your result in part (ii), determine on which of the intervals

(0,1)

and

(1,4)

the inequality

f(x)>g(x)

holds. Give reasons for your answer.