For a real parameter

MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity

Problem

a

, consider the functions

f(x)=e^{x}

and

g_a(x)=ax+2

on the closed interval

[0,2]

. (1) Write an explicit expression for

h_a(x)=f(x)-g_a(x)

. (2) Compute

h_a(0)

and

h_a(2)

in terms of

a

. State the sign of each value when

a=1

a=3

, and

a=5

. (3) Determine all real values of

a

for which the equation

e^{x}=ax+2

has at least one solution

x

in the interval

(0,2)

. Justify your answer thoroughly.