Mathematics

Problem

Define the function

F:\mathbb{R}\to\mathbb{R}

F(x)=\begin{cases} ax^{2}+bx+1, & x<1,\\ 3, & x=1,\\ 2x+a, & x>1, \end{cases}

where

a

and

b

are real parameters. Answer the following. a. Part 1: (i) Obtain the conditions on

a

and

b

required for

F

to be continuous at

x=1

. b. Part 2: (ii) Find the left-hand derivative and the right-hand derivative of

F

x=1

in terms of

a

and

b

. c. Part 3: (iii) Hence determine the values of

a

and

b

for which

F

is differentiable at

x=1

. d. Part 4: (iv) For the values of

a

and

b

obtained in part (iii), write the simplified single-formula expression for

F'(x)

valid for all

x\in\mathbb{R}