Let $g(x)=\begin{cases} \dfrac{px^2-qx+3}{x-1},& x<1,\\ rx+s,& 1\le x<4,\\ x^2-2x+5,& x\ge 4. \end{cases}$ …

Mathematics

Problem

Let

g(x)=\begin{cases} \dfrac{px^2-qx+3}{x-1},& x<1,\\ rx+s,& 1\le x<4,\\ x^2-2x+5,& x\ge 4. \end{cases}

The real numbers

p,q,r,s

are to be chosen so that

g

is continuous at both

x=1

and

x=4

. a. (a) Derive the two independent equations that

p,q,r,s

must satisfy for continuity at

x=1

. b. (b) Derive the two independent equations that

r,s

must satisfy for continuity at

x=4

. c. (c) Hence solve the resulting system to find all quadruples

(p,q,r,s)

for which

g

is continuous at both specified points.