Let f ( x ) = [ x ] 2 - [ x + 3 ] - 3 , x , where [ · ] is the greatest integer function. Then: [2026]

Question

Let $f (x) = {[x]}^{2} - [x + 3] - 3, x \in ℝ,$ where $[\cdot]$ is the greatest integer function. Then: [2026]

Smart Achievers · Accepted Answer

(1)

$f (x) = {[x]}^{2} - [x] - 6 = ([x] + 2) ([x] - 3)$

$(1) f (x) > 0 \Rightarrow [x] \in (- \infty, - 2) \cup (3, \infty)$

$\Rightarrow x \in (- \infty, - 2) \cup [4, \infty)$

$(2) f (x) < 0 \Rightarrow [x] \in (- 2, 3)$

$\Rightarrow x \in [- 1, 3)$

$option (2) is correct$

$(3) \int_{0}^{2} f (x) d x = \int_{0}^{1} (0 - 0 - 6) d x + \int_{1}^{2} (1 - 1 - 6) d x$

$= - 6 - 6$

$= - 12$

$(4) f (x) = 0 \Rightarrow [x] = 3 or [x] = - 2$

$infinitely many solutions$

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