Sets, Relations and Function: Let f(x)=[x]^2-[x+3]-3 , xinmathbfR , where [cdot] is the greatest… | JEE Main 2026

JEE Main 2026 (Online) 22nd January Evening ShiftQuestion 1

Let $f(x)=[x]^2-[x+3]-3$, $x\in\mathbf{R}$, where $[\cdot]$ is the greatest integer function. Then

Sets, Relations and Function

$\int_0^2f(x)\,dx=-6$$f(x)=0$ for finitely many values of $x$$f(x)>0$ only for $x\in[4,\infty)$$f(x)<0$ only for $x\in[-1,3)$✓ Correct answer

Solution

$f=[x]^2-[x]-6=([x]-3)([x]+2)$. $\int_0^2f=-6-6=-12$; zeros on $[3,4)$ and $[-2,-1)$ (infinitely many); $f>0$ also for $x<-2$; $f<0$ iff $[x]\in\{-1,0,1,2\}$, i.e. $x\in[-1,3)$.

Sets, Relations and Function - JEE Mathematics question (JEE Main 2026 (Online) 22nd January Evening Shift)

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