Concept:Simplify f(x) using [x+3]=[x]+3 to get f(x)=[x]2−[x]−6=([x]−3)([x]+2).Explanation:f(x)=0 when [x]=3 or [x]=−2, giving infinitely many x in [3,4) and [−2,−1). Option A is false.f(x)<0 when −2<[x]<3, i.e., [x]∈{−1,0,1,2}, which means x∈[−1,3). Option B is correct.∫02f(x)dx: On [0,1), [x]=0 so f(x)=−6; on [1,2), [x]=1 so f(x)=−6. The integral equals (−6)(1)+(−6)(1)=−12, not −6. Option C is false.f(x)>0 when [x]<−2 or [x]>3, i.e., x<−2 or x≥4. Option D says only x≥4, so false.Only option B holds.Answer:Option B: f(x)<0 only for x∈[−1,3).