Question
Let f(x)=[x]^ 2 -[x+3]-3, x R , where [] is the greatest integer funtion. Then
JEE Main
Mathematics
Definite Integration
2026
JEE Main 2026 (22 January Shift 2)
JEE Main Mathematics Question (2026) — Solution
Options
- A. _ 0 ^ 2 f(x) d x=-6
- B. f(x)<0 only for x [-1,3)
- C. f(x)>0 only for x [4, )
- D. f(x)=0 for finitely many values of x
Answer
B. f(x)<0 only for x [-1,3)
Step-by-step solution
For x [n, n+1) where n is an integer, [x] = n and [x+3] = n+3. Thus f(x) = n^2 - (n+3) - 3 = n^2 - n - 6 = (n-3)(n+2). For f(x) This corresponds to x [-1, 3). We can verify: For n = -1: f = 1 - 2 - 3 = -4 For n = 0,1: f = -6 For n = 2: f = 4 - 5 - 3 = -4 For n = 3: f = 9 - 6 - 3 = 0 For n 4, (n-3)(n+2) > 0. Therefore f(x) < 0 only for x [-1, 3).
Related: Mathematics — Definite Integration · All PYQ Banks