Mathematics

Problem

Consider the functions

g(x)=e^{x}

and

h(x)=x+1

on the closed interval

[0,2]

. Define

F(x)=g(x)-h(x)

. a. Compute

F'(x)

and determine its sign for every

x

(0,2)

. b. Given that

F(0)=0

, use your result from part (a) to decide the sign of

F(x)

for

0<x\le 2

. State what this implies about the relative sizes of

e^{x}

and

x+1

(0,2]

. c. Determine how many solutions the equation

e^{x}=x+1

has in the interval

[0,2]

, and justify your answer thoroughly.