Solve the Greatest Integer Function Equation
Question:
If
\[ [x]^2-5[x]+6=0 \]
where \([\,]\) denotes the greatest integer function, then
(a) \(x\in[3,4]\)
(b) \(x\in(2,3]\)
(c) \(x\in[2,3]\)
(d) \(x\in[2,4)\)
Solution:
Let
\[ [x]=t \]
Then,
\[ t^2-5t+6=0 \]
\[ (t-2)(t-3)=0 \]
\[ t=2 \quad \text{or} \quad t=3 \]
So,
\[ [x]=2 \Rightarrow x\in[2,3) \]
\[ [x]=3 \Rightarrow x\in[3,4) \]
Therefore,
\[ x\in[2,4) \]
\[ \boxed{\text{Correct Answer: (d)}} \]