Mathematics

Problem

Consider the function

f:\\mathbb{R}\to\\mathbb{R}

defined by

f(x)=\begin{cases} px+q, & x<1,\\ x^{2}+1, & 1\le x\le 4,\\ ax+b, & x>4, \end{cases}

where

p,q,a,b

are real constants. In addition, the graph of

f

passes through the points

(0,2)

and

(5,10)

. a. (i) Using the given information, obtain the relation between

p

and

q

that follows from

f(0)=2

. b. (ii) Apply the condition of continuity at

x=1

to form a second equation in

p

and

q

, and hence determine the values of

p

and

q

. c. (iii) Use the condition of continuity at

x=4

to obtain a relation between

a

and

b

. d. (iv) Employ the fact that

f(5)=10

to find the numerical values of

a

and

b

. e. (v) With the constants now fixed, prove that

f

is continuous at every real number.