JEE Mains · Maths · STD 12 - 1. relation and function
Let \( f(x)=[x]^{2}-[x+3]-3, x\in\mathbb{R} \) where \( [\bullet]\) is the greatest integer function. Then
- A \(f(x)>0\) only for \(x \in[4, \infty)\)
- B \(f(x)<0\) only for \(x \in[-1,3)\)
- C \(\int_0^2 f(x) d x=-6\)
- D \(f(x)=0\) for finitely many values of \(x\).
Answer & Solution
Correct Answer
(B) \(f(x)<0\) only for \(x \in[-1,3)\)
Step-by-step Solution
Detailed explanation
\(f(x)=[x]^2-[x]-6=([x]+2)([x]-3)\) (1) \(f(x)>0 \Rightarrow[x] \in(-\infty,-2) \cup(3, \infty)\) \(\Rightarrow x \in(-\infty,-2) \cup[4, \infty)\) (2) \(f(x)<0 \Rightarrow[x] \in(-2,3)\) \(\Rightarrow x \in[-1,3)\) option (2) is correct (3)…
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