Mathematics

Problem

A twice-differentiable function

g

satisfies

g'(x)=p e^{p x}+q,

where

p

and

q

are real constants. Additional information is provided: *

g(0)=2

. *

g(1)=e+2

. * The inverse function

g^{-1}

exists in a neighborhood of

x=0

, and

(g^{-1})'(2)=\dfrac12

. Respond to each part, giving full justification. a. Express

g(x)

in terms of

p

q

, and an integration constant, and then use

g(0)=2

to determine that constant. b. Using

g(1)=e+2

, write an equation relating

p

and

q

. c. Using

(g^{-1})'(2)=\dfrac12

, write a second independent equation relating

p

and

q

. d. Solve the system of equations obtained in parts (b) and (c) to find the exact values of

p

and

q

. Explain how you know your solution is unique.