Mathematics

Problem

Define

F_{a,b}(x)=\frac{x^2+ax+b}{x^2+(a-3)x+(b-3)},

where

a,b\in\mathbb R

. (i) Derive the condition on

a

and

b

that guarantees

F_{a,b}

is defined for every real

x

. (ii) Obtain the conditions on

a

and

b

under which

F_{a,b}

is undefined at exactly two distinct real numbers. (iii) Among the pairs

(a,b)

found in part (ii), identify all those for which the two excluded numbers are consecutive integers.